Sobolev Met Poincaré

MaxPlanckInstitut

fur Mathematik

in den Naturwissenschaften

Leipzig

Sob olev met Poincare

Piotr Hajlasz and Pekka Koskela

PreprintNr

Sob olev met Poincare

Piotr Ha jlasz and Pekka Koskela

Contents

Intro duction

What are Poincare and Sob olev inequalities

Poincare inequalities p ointwise estimates and Sob olev classes

Examples and necessary conditions

Sob olev typ e inequalities by means of Riesz p otentials

Trudinger inequality

A version of the Sob olev emb edding theorem on spheres

RellichKondrachov

Sob olev classes in John domains

John domains

Sob olev typ e inequalities

Poincare inequality examples

Riemannian manifolds

Upp er gradients

Top ological manifolds

Gluing and related constructions

Further examples

CarnotCaratheo dory spaces

CarnotCaratheo dory metric

Upp er gradients and Sob olev spaces

Carnot groups

Hormander condition

Further generalizations

Graphs

Applications to PDE and nonlinear p otential theory

Admissible weights

Sob olev emb edding for p

App endix

Measures

Uniform integrability

p p

spaces L and L

Covering lemma

Maximal function

Leb esgue dierentiation theorem

Abstract

There are several generalizations of the classical theory of Sob olev spaces as

they are necessary for the applications to CarnotCaratheo dory spaces sub el

liptic equations quasiconformal mappings on Carnot groups and more general

Lo ewner spaces analysis on top ological manifolds potential theory on innite

graphs analysis on fractals and the theory of Dirichlet forms

The aim of this pap er is to present a unied approach to the theory of Sob olev

spaces that covers applications to many of those areas The variety of dierent

areas of applications forces a very general setting

We are given a metric space X equipp ed with a doubling measure A

generalization of a Sob olev function and its gradient is a pair u L X

lo c

g L X such that for every ball B X the Poincaretyp e inequality

Z Z

ju u j d Cr g d

B B

holds where r is the radius of B and C are xed constants Working

in the ab ove setting weshow that basically all relevant results from the classical

theory have their counterparts in our general setting These include Sob olev

Poincare typ e emb eddings RellichKondrachov compact emb edding theorem

and even a version of the Sob olev emb edding theorem on spheres The second

part of the pap er is devoted to examples and applications in the ab ovementioned

areas

This research was b egun while PH was visiting the Universities of Helsinki and

Jyvaskyla in continued during his stay in the ICTP in Trieste in and nished

in MaxPlanck Institute MIS in Leipzig in He wishes to thank all the institutes

for their hospitality PH was partially supp orted by KBN grantno POA

PK by the Academy of Finland grant SA and by the NSF

Mathematical Subject Classication Primary E

Intro duction

The theory of Sob olev spaces is a central analytic to ol in the study of various asp ects of

partial dierential equations and calculus of variations However the scop e of its appli

cationsismuch wider including questions in dierential geometry algebraic top ology

complex analysis and in probability theory

Let us recall a denition of the Sob olev spaces Let u L where isanopen

subset of IR and p We say that u belongs to the Sob olev space W if

the distributional derivatives of the rst order b elong to L This denition easily

extends to the setting of Riemannian manifolds as the gradientiswell dened there

The fundamental results in the theory of Sob olev spaces are the socalled Sob olev

emb edding theorem and the RellichKondrachov compact emb edding theorem The

p p

rst theorem states that for p n W L where p npn p

provided the b oundary of is suciently nice The second theorem states that for

p q

every qp the emb edding W L is compact

Since its intro duction the theory and applications of Sob olev spaces have b een under

intensive study Recently there have b een attempts to generalize Sob olev spaces to the

setting of metric spaces equipp ed with a measure Let us indicate some of the problems

that suggest such a generalization

Study of the CarnotCaratheo dory metric generated by a family of vector elds

Theory of quasiconformal mappings on Carnot groups and more general Lo ewner

spaces Analysis on top ological manifolds Potential theory on innite graphs

Analysis on fractals

Let us briey discuss the ab ove examples The CarnotCaratheo dory metric app ears

in the study of hyp o elliptic op erators see Hormander Feerman and Phong

Jerison Nagel Stein and Wainger Rotschild and Stein SanchezCalle

The Sob olev inequality on balls in CarnotCaratheo dory metric plays a crucial role

in the socalled Moser iteration technique used to obtain Harnack inequalities and

Holder continuity for solutions of various quasilinear degenerate equations The pro of

of the Harnack inequalityby means of the Moser technique can b e reduced to verifying

a suitable Sob olev inequality Conversely a parab olic Harnack inequality implies a

version of the Sob olev inequality as shown by SaloCoste It seems that the

rst to use the Moser technique in the setting of the CarnotCaratheo dory metric were

Franchi and Lanconelli The later work on related questions include the pap ers

by Biroli and Mosco Buckley Koskela and Lu Cap ogna Danielli and

Garofalo ChernikovandVodopyanov Danielli Garofalo

Nhieu Franchi Franchi Gallot and Wheeden Franchi Gutierrez and

Wheeden Franchi and Lanconelli Franchi Lu and Wheeden

Franchi and Serapioni Garofalo and Lanconelli Marchi

The theory of CarnotCaratheo dory metrics and related Sob olev inequalities can

b e extended to the setting of Dirichlet forms see Biroli and Mosco Garattini

Sturm

For connections to the theory of harmonic maps see the pap ers Jost

Jost and Xu Ha jlasz and Strzelecki

The theory of quasiconformal mappings on Carnot Groups has b een studied by

Margulis and Mostow Pansu Koranyi and Reimann Heinonen and

Koskela Vo dopyanov and Greshnov Results on Sob olev spaces play an

imp ortant role in this theory Very recently Heinonen and Koskela extended the

theory to the setting of metric spaces that supp ort atyp e of a Sob olev inequality

Semmes has shown that a large class of top ological manifolds admit Sob olev

typ e inequalities see Section Sob olev typ e inequalities on a Riemannian manifold

are of fundamental imp ortance for heat kernel estimates see the survey article of

Coulhon for a nice exp osition

Discretization of manifolds has lead one to dene the gradient on an innite graph

using nite dierences and then to investigate the related Sob olev inequalities see

Kanai Auscher and Coulhon Coulhon Coulhon and Grigoryan

Delmotte Holopainen and Soardi These results have applications

to the classication of Riemannian manifolds Also the study of the geometry of

nitely generated groups leads to Sob olev inequalities on asso ciated Cayley graphs

see Varop oulos SaloCoste and Coulhon and Section for references

At last but not least the Brownian motion on fractals leads to an asso ciated Laplace

op erator and Sob olev typ e functions on fractals see Barlow and Bass Jonsson

Kozlov Kigami Kigami and Lapidus Lapidus

Metz and Sturm Mosco

How do es one then generalize the notion of Sob olev space to the setting of a metric

space There are several p ossible approaches that we briey describ e below

In general the concept of a partial derivative is meaningless on a metric space

However it is natural to call a measurable function g an upp er gradient of a

function u if

jux uy j gds

holds for each pair x y and all rectiable curves joining x y Thus in the Euclidean

setting we consider the length of the gradient of a smo oth function instead of the actual

gradient The ab ove denition is due to Heinonen and Koskela

Assume that the metric space is equipp ed with a measure Then we can ask if for

every pair u g where u is continuous and g is an upp er gradientofutheweak version

Z Z

q p

ju u j d g d Cr

B B

of the Sob olevPoincare inequality holds with qp whenever B is a ball of radius

r Here C and are xed constants barred integrals over a set A mean integral

averages and u is the average value of u over B

If q we call a pPoincare inequality It turns out that a pPoincare inequality

implies a Sob olevPoincare inequalitysee Section

This approachishowever limited to metric spaces that are suciently regular There

have to b e suciently many rectiable curves which excludes fractals and graphs For

more information see Section Section Section Bourdon and Pa jot

Cheeger Franchi Ha jlasz and Koskela Hanson and Heinonen Heinonen

and Koskela Kallunki and Shanmugalingam Laakso Semmes

Shanmugalingam Tyson

Recently Ha jlasz intro duced a notion of a Sob olev space in the setting of an

arbitrary metric space equipp ed with a Borel measure that we next describ e

One can prove that u W p where IR is a b ounded set

with suciently regular b oundary if and only if u L and there is a nonnegative

function g L such that

jux uy j jx y jg x g y

Since this characterization do es not involve the notion of a derivative it can be used

to dene Sob olev space on an arbitrary metric space see Ha jlasz These spaces

have b een investigated or employed in Franchi Ha jlasz and Koskela Franchi Lu

and Wheeden Ha jlasz Ha jlasz and Kinnunen Ha jlasz and Martio

Heinonen Heinonen and Koskela Kalama jska Kilp elainen Kin

nunen and Martio Kinnunen and Martio Koskela and MacManus

Shanmugalingam

Another approach is presented in the pap er of Ha jlasz and Koskela in which

also some of the results from our current work were announced Given a metric space

equipp ed with a Borel measure we assume that a pair u and g g of lo cally

integrable functions satises the family of Poincare inequalities with q and a

xed p onevery ball that is

Z Z

ju u j d C r g d

B P

B B

This family of inequalities is the only relationship between u and g Then we can ask

for the prop erties of u that follow

Yet another approach to Sob olev inequalities on metric spaces is presented in the

pap er by Bobkov and Houdre However it is much dierent from the ab ove men

tioned setting and it will not b e discussed here

One of the purp oses of this pap er is to systematically develop the theory of Sob olev

spaces from inequality This includes the study of the relationships b etween

and Weshow that basically all relevant results from the classical theory have their

counterparts in our general setting These include Sob olevPoincaretyp e emb eddings

RellichKondrachov compact embedding theorem and even a version of the Sob olev

emb edding theorem on spheres

We will work with metric spaces equipp ed with a doubling measure Such spaces

are often called spaces of homogeneous typebutwe will call them doubling spaces The

reader may nd many imp ortant examples of spaces of homogeneous typ e in Christ

and Stein The class of such spaces is pretty large For example Volb erg and

Konyagin proved that every compact subset of IR supp orts a doubling

measure see also Wu and Luukkainen and Saksman

Starting from the work of Coifman and Weiss spaces of homogeneous

typ e have b ecome a standard setting for the harmonic analysis related to singular inte

grals and Hardy spaces see eg Gatto and Vagi Genebahsvili Gogatishvili

Kokilashvili and Krb ec Han Han and Sawyer Macias and Segovia

However it seems that till the last few years there was no development of the theory

of Sob olev spaces in such generality

There are some pap ers on Sob olev inequalities on spaces of homogeneous typ e re

lated to our work see Franchi Lu and Wheeden Franchi Perez and Wheeden

MacManus and Perez the last three pap ers were motivated by our

approach Also the pap er of Garofalo and Nhieu provides a similar approach in

the sp eial case of CarnotCaratheo dory spaces

The reader might wonder why we insist on studying the situation with a xed

exp onent p instead of assuming that holds with p There is a simple reason

for this Indeed for each p one can construct examples of situations where

holds for each smo oth function u with g jruj but where one cannot replace p byany

exp onent q p Let us give an example to illustrate the dep endence on p Take two

threedimensional planes in IR whose intersection is a line L and let X be the union

of these two planes The metrics and measures induced from the planes have natural

extensions to a metric and a measure on X If u is a smo oth function on X then we

dene g x to be jruxj whenever x do es not belong to L where ru is the usual

gradient of u in the appropriate plane and dene g x to be the sum of the lengths

of the two gradients corresp onding to the dierent planes when x L One can then

check that holds for p but fails for p

As we said we want to develop the theory of Sob olev spaces assuming a family of

Poincare inequalities and the doubling prop erty Such an approach has found many

applications in the literature in various areas of analysis and geometry The applications

include the ab ove mentioned CarnotCaratheo dory spaces graphs

Dirichlet forms quasiconformal mappings

This approach has also found many imp ortant applications in Riemannian geometry

The class of op en Riemannian manifolds that satisfy b oth the doubling condition and

the pPoincare inequality is under intensive investigation see Colding and Minicozzi

Grigoryan Holopainen Holopainen and Rickman Li and

Wang Maheux and SaloCoste Rigoli Salvatori and Vignati Salo

Coste Tam where some global prop erties of manifolds were

obtained under the assumption that the Riemannian manifold satises a pPoincare

inequality and the doubling prop erty

In Yau proved that on op en Riemannian manifolds with nonnegative

Ricci curvature b ounded harmonic functions are constant Some time later he conjec

tured that for such manifolds space of harmonic functions with p olynomial growth of

xed rate is nite dimensional

Indep endently Grigoryan and SaloCoste generalized Yaus theorem

byproving that b ounded harmonic functions are constant provided the manifold satis

es the doubling prop erty and the Poincare inequalitywithg jruj and p It

is known that manifolds with nonnegative Ricci curvature satisfy those two conditions

see Section Under the same assumptions the result of Yau has been extended to

harmonic mappings see Li and Wang and Tam

Very recently Colding and Minicozzi answered the conjecture of Yau in

the armative Again the assumptions were that the manifold is doubling and that

the Poincare holds

Many of the ab ove Riemannian results have counterparts in the more general set

tings of CarnotCaratheo dory spaces graphs or Dirichlet forms and again the main

common assumption is the same doubling and Poincare

This common feature was guiding us in our work The rst part of the pap er is

devoted to general theory and the second part to examples and applications in the

areas mentioned ab ove

The pap er is organized as follows In Section we present the setting in which

we later on develop the theory of Sob olev inequalities In Section we discuss the

equivalence of various approaches to Sob olev inequalities on metric spaces Section is

devoted to some basic examples and conditions that are necessarily satised by spaces

that satisfy all pPoincare inequalities for pairs of a continuous functions and upp er

gradients In Section we show that if a pair u g satises a pPoincare inequality

then ju u j can be estimated by a generalized Riesz p otential This together with a

generalization of the Fractional Integration Theorem implies a variant of the Sob olev

Poincare emb edding theorem In Section we imp ose the additional condition that

the space be connected and improveon one of the inequalities from Section namely

we prove a variant of the Trudinger inequality In Section we prove an emb edding

theorem for almost all spheres centered at a given point In Section we generalize

the classical RellichKondrachov theorem to the setting of metric spaces So far all

the results are lo cal in nature In Section we intro duce the class of John domains

and generalize previous results as global results in John domains In Section we

collect imp ortant examples of metric spaces where the theory develop ed in the pap er is

applicable including op en Riemannian manifolds top ological manifolds and Lo ewner

spaces In Section we study the theory of CarnotCaratheo dory spaces that are

asso ciated with a family of vector elds from the p oint of view of Sob olev inequalities

on metric spaces In Section we discuss Sob olev inequalities on innite graphs

Section is devoted to applications of the theory to nonlinear p otential theory and

degenerate elliptic equations Section is an app endix where we collect all the results

ab out measure theory and maximal functions that are needed in the pap er

The exp osition is selfcontained and the background material needed is the abstract

measure theory in metric spaces some real analysis related to maximal functions and

the basic theory of classical Sob olev spaces covered byeach of the following references

Evans and Gariepy Gilbarg and Trudinger Maly and Ziemer Ziemer

Some examples and applications that illustrate the theory require slightly more In

Section some familiarity with Lie groups and commutators of vector elds is

needed and in Section we assume basic facts ab out quasilinear elliptic equations in

divergence form One can however skip reading Sections and and it will not

aect understanding of the remaining parts of the pap er

We did make some eort to give comprehensive references to sub jects related to our

work We are however sure that many imp ortant references are still missing and we

want to ap ologize to those whose contribution is not mentioned

Notation Throughout the pap er X will be a metric space with a metric d and a

Borel measure The precise assumptions on are collected in the app endix If not

otherwise stated will b e doubling which means that

B C B

whenever B is a ball and B is the ball with the same center as B and with radius twice

that of B in the same waywe dene B for Wewillcallsuch a metric measure

space X a doubling space and C a doubling constant X will always denote an

op en subset Sometimes we will need the doubling prop erty on a subset of X only we

will say that the measure is doubling on if holds whenever B B x r x

and r diam By writing v L we designate that v b elongs to the class

lo c

L B with resp ect to for each ball B If X we will simply write v L

lo c

By Lip X we denote the class of Lipschitz functions on the metric space X

R R

The average value will be denoted by v vd If R and vd A

A A

v is a measurable function M v stands for the restricted HardyLittlewood maximal

function

M v x sup jv j d

B xr

r R

If R we will simply write Mv Another version of the maximal function is

juj d M v x sup

B x r

B xr

which applies to v L It is also clear how to dene the restricted maximal

lo c

function M v

By H we denote the k dimensional Hausdor measure The symbol denotes

the characteristic function of a set E We reserve B to always denote a ball Observe

that according to the structure of the metric space it may happ en that the center and

the radius of the ball are not uniquely dened In what follows when we write B we

assume that the center and the radius are xed Otherwise B is not prop erly dened

By C we will denote a general constant which can change even in a single string of

estimates By writing C C p q we indicate that the constant C dep ends on p q

and only We write u v to state that there exist two p ositive constants C and C

such that C u v C u

Some further notation and commonly used results are collected in the app endix

What are Poincare and Sob olev inequalities

In this section we describ e the general framework and give samples of problems which

are treated later on Until the end of the section we assume that is a Borel measure

on a metric space X but we do not assume that is doubling As b efore X

denotes an op en set

Denition Assume that u L and a measurable function g satisfy the

lo c

inequality

Z Z

ju u j d C r g d

B P

B B

on each ball B with B where r is the radius of B and p C are

xed constants We then say that the pair u g satises a pPoincare inequality in

If X we simply say that the pair u g satises a pPoincare inequality

Note that if u Lip IR g jruj and p then is a corollary of the classical

Poincare inequality

Z Z

p p

ju u j dx C n pr jruj dx

B B

Quite often we will call an inequality weak if b oth sides involve a ball and the radius

of the ball on the right hand side is greater than the radius of the ball on the left hand

side likein

Unfortunately it is easy to see that in general inequality do es not hold with

p cf p Nevertheless there are many imp ortant situations where the

pPoincare inequalities and hold with p For example they hold when u is

a solution to an elliptic equation of a certain typ e see Section For this reason we

include the case p

It is natural to regard a pair u g that satises a pPoincare inequality in as a

Sob olev function and its gradient In this sense we will develop the theory of Sob olev

functions on metric spaces with gradient in L for all p

In the classical approach the Sob olev spaces are dened for p only Moreover

it was exp ected that there were no reasonable theory of Sob olev spaces for p

see Peetre However we obtain a rich theory of Sob olev spaces for all p In

the Euclidean setting when p our approach is equivalent to the classical one

In the literature there are a few pap ers that deal with the Sob olev inequalities for

p see Bakry Coulhon Ledoux and SaloCoste Buckley and Koskela

Buckley Koskela and Lu Calderon and Scott Ha jlasz and Koskela

Let us assume that a pair u g satises a pPoincare inequalityfor pinanopen

set X We inquire for prop erties of u that follow from this assumption A typical

question is whether the Sob olev emb edding theorem holds ie whether the pPoincare

inequality in implies the global Sob olevtyp e inequality

Z Z

p q

C inf g d ju cj d

cIR

with an exp onent q p We suggest the reader to lo ok at our earlier pap er

where a result of this typ e was obtained by an elementary metho d In the current

pap er we obtain stronger results by more complicated metho ds

Note that if and q then the ab ove inequality is equivalent to

Z Z

q p

C ju u j d g d

as for q and we have

Z Z Z

q q q

ju cj d ju cj d ju u j d inf inf

cIR cIR

The classical gradient of a Lipschitz function has a very imp ortant prop erty if the

function is constant in a set E then the gradient equals zero ae in E To have a

counterpart of this prop erty in the metric setting weintro duce the truncation property

Given a function v and t t we set

v minfmaxfv t gt t g

Denition Let the pair u g satisfy a pPoincare inequality in Assume that for

every b IR t t and f g the pair v g where

ft v t g

v u b satises the pPoincare inequality in with xed constants C

Then we say that the pair u g has the truncation property

Let p and u Lip IR Since v u b satises jrv j jruj

ft v t g

ae the pair u jruj has the truncation prop erty More sophisticated examples are

given in Section

We close the section with a result which shows that inequality is equivalent to

aweaker inequalityprovided the pair u g has the truncation prop erty The result will

be used in the sequel

Theorem Let X beanopen set with Fix q p C

and Assume that every pair u g that satises a pPoincare inequality in

with given C and satises also the global MarcinkiewiczSobolev inequality

q p

inf sup fx jux cj tgt C g d

cIR

Then every pair that satises the pPoincare inequality in with given C and and

has the truncation property satises also the global Sobolev inequality

Z Z

q p

inf ju cj d C g d

cIR

with C C

Remarks We call a MarcinkiewiczSob olev inequality because it implies that

u b elongs to the Marcinkiewicz space L

The result is surprising even in the Euclidean case inequality seems much

q q

weaker than as the inclusion L L is prop er Similar phenomena have b een

discovered by V G Mazya cf Section Theorem who

proved that a Sob olev emb edding is equivalent to a capacitary estimate which is a

version of inequality The main idea of Mazya was a truncation method which is

also the key argument in our pro of This metho d mimics the pro of of the equivalence

of the Sob olev inequality with the isop erimetric inequality Inequalityplays a role

of the relative isop erimetric inequality and the truncation argumentprovides a discrete

counterpart of the coarea formula The truncation metho d of Mazya has b ecome

very useful in proving various versions of the Sob olev emb edding theorem with sharp

exp onents in the b orderline case where interp olation arguments do not work To see

how the argument works in the case of the classical Sob olev embedding theorem we

refer the reader to the comments after the statement of Theorem Recently the

truncation metho d has been employed and even rediscovered by many authors see

Adams and Hedb erg Theorem Bakry Coulhon Ledoux and SaloCoste

Biroli and Mosco Cap ogna Danielli and Garofalo Franchi Gallot and

Wheeden Garofalo and Nhieu Heinonen and Koskela Long and

Nie Maheux and SaloCoste Semmes and Tartar

Proof of Theorem Let u g be a pair which satises the pPoincare inequality

in and which has the truncation prop erty Cho ose b IR such that

fu bg and fu bg

q q

Let v maxfu b g v minfu b g We will estimate kv k and kv k

L L

separately In what follows v will denote either v or v

Lemma Let be a nite measureonasetY If w is a measurable function

such that fw g Y then for every t

g fwtg inf fjw cj

cIR

The pro of of the lemma is easy and left to the reader

g satises the pPoincare in By the truncation prop erty the pair v

ft v t g

equality and hence it satises Moreover the function v has the prop erty

g Hence applying the lemma we conclude that fv

t t

q q t t

cj tgt inf sup jv sup fv

t t

cIR

t t

C kg k

ft v t g

This yields

q kq k k

v d f v g

kq k

fv g

k kq

g fv

p q

g k k d C

f v g

p q

g k k d C

f v g

C kg k

In the second to the last step we used the inequality qp Finally

Z Z Z

q q q

q q

ju bj v v C kg k

This completes the pro of

The following theorem is a mo dication of the ab ove result

Theorem Let q p C and Assume that every pair u

g that satises the pPoincare inequality with given C and satises also the weak

MarcinkiewiczSobolev inequality

fx B jux cj tgt

p q

g d C r inf sup

cIR

for every bal l B where r denotes the radius of B Then every pair u g that satises the

pPoincare inequality with given C and and has the truncation property satises

also the weak Sobolev inequality

Z Z

q p

inf ju cj d C r g d

cIR

B B

for every bal l B with C C

The pro of is essentially the same as that for Theorem and weleave it to the reader

Poincare inequalities p ointwise estimates and

Sob olev classes

Our starting point to the theory of Sob olev spaces on metric spaces is to assume that

the pair u g satises a pPoincare inequality There are however also other p ossible

approaches Recently Ha jlasz intro duced a notion of a Sob olev space in the

setting of metric space equipp ed with a Borel measure In this section we will compare

this approach to that based on Poincare inequalities see Theorem The pro of

is based on pointwise inequalities which have their indep endent interest and which we

state in a more general version than is needed for the sake of the pro of see Theorem

and Theorem Finally we compare the class of L pairs of u g that satisfy a p

Poincare inequalityto the classical Sob olev space

For a detailed study on the equivalence of various approaches to Sob olev inequalities

on metric spaces see Franchi Ha jlasz and Koskela and Koskela and MacManus

Results related to those of this section app ear also in Franchi Lu and Wheeden

Ha jlasz and Kinnunen Heinonen and Koskela Shanmugalingam

Given p and a triple X d where X d is a metric space and is a

Borel measure not necessarily doubling Ha jlasz denes the Sob olev space

p p p

M X d as the set of all u L X for which there exists g L X such

that

jux uy j dx y g x g y ae

When we say that an inequality like holds ae we mean that there exists a set

E X with E such that inequality holds for all x y X n E

p p

If p the space is equipp ed with a Banach norm kuk kuk inf kg k

L g L

where the inmum is taken over the set of all g L X that satisfy

The motivation for the ab ove denition comes from the following result

n n

If IR or if IR is a b ounded domain with suciently regular b oundary

jj is the Euclidean metric H the Leb esgue measure and p then

p p n

W M jjH

as sets and the norms are equivalent see and also Here W

denotes the classical Sob olev space of L integrable functions with generalized gradient

in L If p the equivalence fails see and also However for any

p n p

op en set IR and p M jjH W see Prop osition

and also Lemma

For the further development and applications of the ab ove approach to Sob olev

spaces on metric space see Franchi Ha jlasz and Koskela Franchi Lu and Whee

den Ha jlasz and Kinnunen Ha jlasz and Martio Heinonen

Heinonen and Koskela Kalama jska Kilp elainen Kinnunen and Martio

Kinnunen and Martio Koskela and MacManus Shanmugalingam

Prior to the work of Ha jlasz Varop oulos dened a function space on a smo oth

compact manifold based on an inequality similar to Recently and indep endently

Vo dopyanov used inequality to dene a Sob olev space on a Carnot group

The following result compares the ab ove denition of the Sob olev space with the

approach based on Poincare inequalities

Theorem Let X be a doubling space If p then the fol lowing conditions

are equivalent

u M X d

p p

u L X and there exist C g L X and q p such

that the Poincare inequality

Z Z

ju u j d Cr g d

B B

holds on every bal l B of radius r

Remarks In fact we prove the implication for any p Under

much more restrictive assumptions ab out the measure Theorem has been proved

by Franchi Lu and Wheeden see also

Proof of Theorem Integrating inequality over a ball with resp ect to x and

y we obtain

Z Z

ju u j d Cr gd

B B

whichproves the implication The opp osite implication follows from Theorem

and the Maximal Theorem

Theorem Let X be a doubling space Assume that the pair u g satises a p

Poincare inequality p Then

p p p p

jux uy j Cdx y M g x M g y

dxy dxy

for almost every x y X where M v x sup jv j d

r R

B xr

Before we prove Theorem we show how to use it to complete the pro of of the

implication Assume that u g L X satisfy Then inequality holds

with p replaced by q Note that

q q

x Mg M g x

dxy

q pq q q p

Now g L pq and so the Maximal Theorem implies Mg L

and hence the claim follows

Proof of Theorem Let x y X be Leb esgue points of u by the Leb esgue

dierentiation theorem see Theorem this is true for almost all p oints Write

B xB x r B x dx y for each nonnegative integer i Then u ux

i i B x

as i tends to innity Using the triangle inequality the doubling of and the pPoincare

inequality we conclude that

jux u j ju u j

B x B x B x

i i

ju u j d

B x

C ju u j d

B x

C r g d

B x

p p

C r M g x

i dxy

p p

Cdx y M g x

dxy

Similarly

p p

juy u j Cdx y M g y

B y dxy

Moreover

ju u j ju u j ju u j

B x B y B x B x B y B x

C ju u j d

B x

Cdx y g d

B x

p p

Cdx y M g x

dxy

The claim follows by combining the ab ove three inequalities This completes the pro of

of Theorem and hence that for Theorem

It is interesting to observe that Theorem can be converted see also Heinonen

and Koskela This is the content of the following result

Theorem Let X be a doubling space and u L X g L X

lo c

p Suppose that the pointwise inequality

p p p p

jux uy j Cdx y M g x M g y

dxy dxy

holds for almost al l x y X with some xed Then the pPoincare inequality

Z Z

ju u j d C r g d

B P

B B

holds for al l bal ls B Here C depends only on p C C

P d

Proof Fix aballB with the radius r Then for almost all x y B we have

p p p p

jux uy j Cdx y M g x M g y

B B

Fix t Taking an average with resp ect to x and y applying Cavalieris princi

ple see Theorem and the weak typ e estimate for the maximal function see

Theorem we obtain

Z Z

ju u j d Cr M g d

B B

p p

CrB fx B M g t g dt

Z Z Z

CrB B dt g d dt

t B

CrB t B Ct g d

p p

g d The pro of is complete The claim follows when we cho ose t B

Note that the argument used ab ove is similar to that used in the pro of of Theo

rem

Theorem suggests the following question Is it true that if a pair u g L X

p satises a pPoincare inequality in a doubling space X then there exists

qp such that the pair u g satises a q Poincare inequality This seems to be a

very delicate question see the discussion in the remark in Section below

If the answer to the ab ove question were armative Theorem would imply a

p p p

stronger result u M p if and only if u L and there is g L X such

that the pair u g satises a pPoincare inequality

In the sp ecial case when X IR d is the Euclidean metric and is the Leb esgue

measure the answer to the ab ove question is in the p ositive due to the results of

Franchi Ha jlasz and Koskela and Koskela and MacManus

The following theorem was proved in The result is a generalization of some

results in

Theorem Let u g L IR g p Suppose that there exist and C

such that

Z Z

ju u j dx Cr g dx

B B

n n

for al l bal ls B IR Then u W IR and jruj C g ae In particular

Z Z

ju u j dx C r gdx

B B

for al l bal ls B IR

Note that it follows from the results in this section that if a pair u g L satises a

for any q p Indeed Theorem pPoincare inequality p then u W

lo c

together with the weak typ e estimate for the maximal function and the emb edding

q q

L L for all qpsee Theorem imply that for some h L the inequality

lo c lo c

jux uuj jx y jhxhy holds ae Then the claim follows from This

argument however do es not guarantee that u W IR

For far reaching generalizations of Theorem see and and also Sec

tion

Examples and necessary conditions

We rst discuss three examples that indicate the dep endence of the validity of p

Poincare inequalities on the exp onent p Notice that if a pair u g satises a pPoincare

inequality then it satises a q Poincare inequality for all q p by Holders inequality

The following examples show that this is not the case for qp

Example Let X R be equipp ed with the Euclidean metric and let be the

measure generated by the density dx jx j dx t where x denotes the second

co ordinate of x Then the Poincare inequality holds for each Lipschitz function u

with g jruj if and only if p t

The Poincare inequality holds as is an A weight for the indicated values of p see

Theorem On the other hand the pPoincare inequality

fails for p t and hence for p t To see this let B be the disk of radius

and with center Let us consider a sequence u of Lipschitz functions that only

dep end on x and such that u if x B and x u if x B and x

i i

u xi log x if x Then

Z Z

t t

jru j d i log

which tends to zero as i approaches innity On the other hand ju x u j for

i i

all x either in the part of B ab ove the line x or in the part b elow the line x

Hence the integral of ju u j over B is b ounded away from zero indep endently of i

i i

and so the tPoincare inequality cannot hold for all Lipschitz functions Using a

standard regularization argument we can then assume that functions u in the ab ove

example are C smo oth so the tPoincare inequality cannot hold for all C

smo oth functions either

Example Let X fx x x IR x x x g be equipp ed

n n

with the Euclidean metric of IR and with the Leb esgue measure The set X consists

of two innite closed cones with a common vertex Denote the upp er cone by X and

the lower one by X

WewillprovethatthepPoincare inequality holds in X for every pair u g where

u is a continuous function and g an upp er gradientofu ifandonlyifpn For more

information ab out upp er gradients see Section

First we prove that the inequality fails for p n and hence for p n Fix

n n

Since x log j log jxjj satises W B and x as

x we can truncate the function and obtain a continuous function u W X

such that u u x for jxj and kru k We extend this

L X

function to the lower cone as the constant Fix a ball B centered at the origin Then

kru k while ku u k C uniformly with resp ect to and thus the

L B

nPoincare inequalitycannot hold

It remains to prove the inequality for pn Since jruj g for an upp er gradient

of u Prop osition it suces to prove the pPoincare inequality for the pair u

jruj By Theorem it suces to verify the pointwise estimate

p p p

jux uy j C jx y jM jruj xM jruj y

jxy j jxy j

We can assume that x and y belong to dierent cones as the pPoincare inequality

holds in each of those two cones Then by the triangle inequality either jux uj

jux uy jor juy uj jux uy j Assume that the rst inequality holds

B x jxj Then by the emb edding into Holder continuous functions Let X

np p p p p

jux uj C jx j jruj C jx y jM jruj x

jxy j

This ends the pro of of the claim

Mo difying the argument used ab ove one can construct many other examples Let

for example X be the union of two dimensional planes in IR whose intersection is a

line Equip X with the dimensional Leb esgue measure and with the metric induced

by the Euclidean metrics of the planes Then the pPoincare inequality holds in X for

all pairs u g where u is a continuous function in X and g an upp er gradient of u if

and only if p

Amuch more general result that allows one to build similar examples in the setting

of metric spaces was proven by Heinonen and Koskela Theorem

Example For each p n there is an op en set X IR equipp ed with the

Euclidean metric and the the Leb esgue measure such that the pPoincare inequality

holds for each smo oth function u with g jruj but no Poincare inequality holds

for smaller exp onents for all smo oth functions

Such an example was constructed by Koskela We will recall the idea of the

n n

example following Let E IR be a compact set such that W IR n E

n n n

p q q

W IR and W IR is a prop er subset of W IR n E for all qp In other

p q

words the set E is W removable but it is not W removable for any q p

Such sets were explicitly constructed in In fact there is a smo oth function u in

n n

IR n E such that jruj L IR for all qp but the pair u jruj do es not satisfy a q

Poincare inequalityinIR n E for any qp The pair satises the pPoincare inequality

n n n

p p p

but jruj L IR n E as otherwise we would have u W IR n E W IR

and hence it would even satisfy the Poincare inequality Thus this example do es not

solve the question p osed after Theorem

Remark The last example shows that it mayhappenthatapPoincare inequality holds

for all smo oth pairs u jruj in X and there is a smo oth function u in X such that no

q Poincare inequality holds for any qpfor the pair u jruj However in this example

jruj L X and hence we do not know if a pPoincare inequality for the pair u g

with g L X implies a q Poincare inequality for some qp As Example shows

a q Poincare inequality cannot hold for all q p in general but we do not know if it

can hold for some qp suciently close to p

Let us next describ e some necessary geometric conditions for the validityofPoincare

inequalities

Denition We say that X is weakly local ly quasiconvex if X is path connected and

for each x X there is a neighborhood U and a continuous function with

such that any pair x x of points in U can be joined in X with a curve of length

no more than dx x If any pair x x of points in X can be joined by a curve

whose length is no more than Cdx x we say that X is quasiconvex

Prop osition Suppose that X is weakly local ly quasiconvex and doubling Let p

If for each pair u g of a continuous function and its upper gradient we have a p

Poincare inequality with xed C then X is quasiconvex

Proof Given a p oint x X dene ux inf l where the inmum is taken over

all rectiable curves joining x x As X is weakly lo cally quasiconvex u is continuous

Moreover it is easy to see that the constant function g is an upp er gradient of u

Thus by Theorem we have

p p p

jux uy j Cdx y M g xM g y Cdx y

dxy dxy

The claim follows from this inequality

Thus the validityofa pPoincare inequality guarantees the existence of short curves

If the doubling measure b ehaves as the Euclidean volume and the exp onent p is no

more than the growth order of the volume then X cannot have narrow parts This

conclusion is a consequence of Prop osition b elow Under the additional assumption

that each closed ball in X be compact this result can be deduced from the results in

Prop osition Suppose that X is weakly local ly quasiconvex and that B x r

r with s for each x and al l r Assume that for each pair u g of a continuous

function and its upper gradient we have an sPoincare inequality with xed C

If x X r and x x B x r n B x r then x x can be joined in

B x Cr n B x rC by a curve whose length does not exceed Cdx x

Notice that the claim of the prop osition would still be true if we replaced the s

Poincare inequalitybyapPoincare inequality p sasapPoincare inequality implies

an sPoincare inequalityby means of the Holder inequality However we cannot replace

the sPoincare inequality by a pPoincare for any ps as follows from Example

Proof The pro of is very similar to the arguments used in the pro of of Corol

lary and in the pro of of Prop osition Throughout C denotes a constant

whose value can change from line to line but that only dep ends on the given data

By Prop osition we may assume that dx x C r Supp ose that x x cannot

be joined in B x Cr n B x rC with C By Prop osition we then obtain

rectiable curves F F B x r n B x r b oth of length no more than Cr but

at least C r and so that dF F C r It suces to showthatF F can b e joined

by acurve of length less than Cr inside B x Cr n B x rC

If follows from the sPoincare inequality and from the volume growth condition that

g d C

B x C r

for any upp er gradient g of any continuous function u that takes on the constantvalue

inF and takes on avalue greater or equal to at each pointof F Indeed assume

rst that ju j Then slightly mo difying the pro of of we get for all

B x r

x F

s s

jux u j Cr sup R g d

B x r

B xR

Thus for some R r

C R r g d

B xR

Now inequality follows from the covering lemma Theorem and the fact that

S P

if F B r then r C r If ju j then inequality follows by

i i i

B x r

a symmetric argument The pro of of is complete

Now set g x log dx x in B x Cr n B x C r and extend g as

zero to the rest of X Supp ose that F F cannot b e joined in B x Cr n B x C r

by a rectiable curve Dene u xinf g ds where the inmum is taken over all

rectiable curves that join x to F Then g is an upp er gradient of u the restriction

of u to F is zero and ux at each point of F By the preceding paragraph

we see that the integral of g over B x Cr is b ounded away from zero and hence a

computation using the volume growth condition b ounds the constant C in the denition

of g from ab ove Thus wecanxC so that F F can b e joined by a rectiable curve

in B x Cr n B x C r

Set a inf l where the inmum is taken over all rectiable curves that join

F to F in B x Cr n B x C r We dene a function u similarly as we dened

u ab ove using g g where g xa x and is the characteristic function

U U

of B x Cr n B x C r As we can make the integral of g as small as we wish by

cho osing the constant C in the denition of g large enough the reasoning used in the

preceding paragraph shows that the integral of g over B x Cr must be b ounded

away from zero and thus the volume growth condition implies that a Cr as desired

Sob olev typ e inequalities by means of Riesz po

tentials

As it was pointed out in Section one of the aims of this pap er is to prove a global

Sob olev inequality

Z Z

q p

inf ju cj d C g d

cIR

where qp or at least a weak lo cal Sob olev inequality

Z Z

q p

Cr inf ju cj d g d

cIR

B B

where and B is any ball of radius r assuming that the pair u g satises a

pPoincare inequalityonly

Inequality requires some additional information on while turns out to

be true in avery general setting

Another question we deal with is how to determine the b est p ossible Sob olev exp o

nent q in the ab ove inequalities and

In the remaining part of the section we will be concerned with inequalities of the

typ e The case of the global Sob olev inequality will b e treated in Section

Let X b e a doubling space Beside the doubling condition we will sometimes require

that

B r

whenever B is an arbitrary ball of radius r and B B x r x B r r

Notice that the doubling condition on always implies for some exp onent s

that only dep ends on the doubling constant of This follows by a standard iteration

of the doubling condition see Lemma in the app endix Inequality could well

hold with exp onents smaller than the one following from the doubling condition and in

the following results s refers to any exp onent for which is valid

Theorem Assume that the pair u g satises a pPoincare inequality p

in a doubling space X Assume that the measure satises condition

If ps then

fx B jux u j tgt

p p

Cr g d

where p sps p and B is any bal l of radius r Hence for every hp

Z Z

h p

ju u j d Cr g d

B B

Moreover for every q such that p q s

Z Z

q q

ju u j d Cr g d

B B

where q sq s q and B is any bal l of radius r If in addition the pair u g

has the truncation property then

Z Z

p p

ju u j d Cr g d

B B

If p s then

C B ju u j

d C exp

r kg k

L B

If p s then u after redenition in a set of measure zero is local ly Holder

continuous and

sup jux u jCr g d

In particular

p sp

g d jux uy j Cr dx y

for al l x y B where B is an arbitrary bal l of radius r

The constants in the theorem depend on p q h s C C and C only

d P b

Remarks Inequality holds also for functions on graphs see Theorem

Assuming that the space is connected we can improve on inequality see Sec

tion

Instead of assuming that X be doubling we could assume for instance that the

doubling condition holds on all balls with radii b ounded from ab ove by r such a

situation o ccurs for example on Riemannian manifolds with a lower b ound on the Ricci

curvature see Section or that it holds on a given op en set Then the inequalities of

the theorem would hold on balls with radii b ounded from ab ove or on small balls cen

tered at the op en set We leave it to the reader to check that the pro of of Theorem

gives such a statement

A mo dication of the pro of shows that the ball B can b e replaced by B

the details are left to the reader

We present only one of the p ossible pro ofs of the ab ove theorem The pro of can also

b e based on the emb edding theorem for Sob olev spaces on metric spaces from Ha jlasz

This approach uses the observation that a family of Poincare inequalities leads

to p ointwise inequalities we do not provide the details here

Since the pro of of the theorem is rather complicated we b egin with some comments

that will explain the idea

p p

In one of the pro ofs of the classical Sob olev emb edding W B L B where

p n p npn p and B is an ndimensional Euclidean ball one rst proves

the inequality

jux u jCI jrujx

B n

where I g x g z jx z j dz and then applies the Fractional Integration Theo

rem which states that

B p p

I L B L B

is abounded op erator for pn If p one only gets aweak typ e estimate

jfx B I g x tgjt C jg z j dz

in place of which in turn leads to the emb edding W B L B Then

the emb edding W B L B follows from Theorem

The main idea of our pro of of inequalities like or is to mimic the ab ove

argument Thus the pro of splits into two steps

Assume that a pair u g satises a pPoincare inequalityinagiven doubling space

In the rst step we prove the inequality

g ju u j CJ

where J is a suitable generalization of the Riesz p otential I and then in the second

step we prove a version of the Fractional Integration Theorem for the op erator J

This will complete the pro of of The pro of of will require a more sophisticated

version of the inequality the details will be completed in Section where we

intro duce an appropriate class of domains for the Sob olevPoincareemb edding

Any inequalityofthe typ e will b e called a representation formula

Before we dene J we start with a discussion on Riesz potentials to explain the

motivation The classical Riesz p otential is dened as

g y

I g x dy

jx y j

where n and is a suitable constant In this pap er the exact value of the

constant is irrelevant to us Moreover for our purp oses any op erator J such that

C I g Jg C I g for g

is as go o d as I

A natural generalization of the Riesz p otential to the setting of doubling spaces is

g y d x y

dy I g x

B x dx y

or its lo cal version

g y d x y

dy I g x

B x dx y

Wewould like to estimate ju u j by CI g but in general this is not p ossible Instead

of that we have to consider a p otential which is strictly larger than I g

Observe that the potential dened by

B x jg y j dy I g x

A x

i i

where A xB x n B xB x n B x is equivalentto I g in the sense

i i i

of Note that if diam then A x so all the summands in

for diam vanish Thus replacing the integral over A x by the integral

over B x and then taking the sum over diam we obtain the new potential

jg j d J g x

B x

diam

which satises I g J g Now we dene

i p

J g x jg j d

B x

diam

where p and are xed constants

The other generalization is

p p

jg y j d x y

dy I g x

B x dx y

A x

Observe that I g I jg jand I g CJ g ae Thus once we prove the fractional

p p

integration theorem for J g it is true for I g as well

p p

In Section we will obtain a version of the representation formula with I g

in place of J see Theorem

Theorem Let the pair u g satisfy a pPoincareinequality in a doubling space X

Then for every bal l B X the representation formula

jux u jCJ g x

holds almost everywhere in B

This representation formula together with a suitable Fractional Integration Theorem

see Theorem will lead to emb edding

Proof The argument is very similar to that used in the pro of of inequality

Let x B be a Leb esgue p oint of u Put D x B x Let i be the least

integer such that diam B Then B D x Since u ux as i

D x

we obtain

jux u j ju u j ju u j

B B

D x D x D x

i i i

i p

C g d

D x

CJ g x

Theorem Let X beanopen and bounded set and let p

Assume that the measure is doubling on V fx X dist x diam g

Moreover assume that for some constants C s

B x r C

diam

whenever x and r diam and that g L V

If p s then J g L where p sps p Moreover

p w

p p p p

fx J gtg C t diam kg k

L V

for t and hence for every r p

r p

kJ g k C diam kg k

L L V

Here the constants C and C depend on p C s and C only

b d

If pq and q s then

kJ g k C diam kg k

L V

where q sq s q and C C pqbsC

If p s then

C B J g

exp d C

diam kg k

L V

where C C pbsC i

i d

If p s then J L and

g k C diam kg k kJ

L L V

where C C pbsC

Proof of Theorem We mo dify a standard pro of for the usual Riesz potentials

All the constants C app earing in the pro of dep end on p q b sand C only

Case p s Take arbitrary q p such that q s Fix r diam

r r

Decomp ose the sum which denes J g into J g J g where J g and J g

r r

For x we have

r diam

p p p i

J g x M jg j x r M jg j x

r V V

Here M h denotes the maximal function relativeto the op en set V

To estimate J g we apply the lower b ound on

r i p

J g x jg j d

B x

r diam

i q q

B x jg j d

B x

r diam

isq sq q q

C diam jg j d

r diam

sq sq q q

Cr diam jg j d

In the last step we used the fact sq to estimate the sum of the series by its

rst summand Now

p sq sq q q

g x C r M jg j r diam J jg j d

Note that

p sq sq q q

r M jg j r diam jg j d

if and only if

p p s

kg k M jg j r diam

L V

If the RHS in do es not exceed diam then we take r equal to the RHS In this

case we get

q s

s p sq sp

g x C diam kg k M jg j x J

L V

and hence

q psq

spsq spsq psq p

J g x C diam kg k M jg j x

L V

If the RHS in is greater than diam then we take r diam Then

J g x C diam kg k

L V

Let A denote the set of p oints in for which holds and let A consist of those

points in that satisfy Write fx J gtg Then

A A

t t t

If we take q p then inequality follows from estimates and

the weak typ e estimate for the maximal function M jg j see Theorem gives

p sp

p spsp

A CDkg k M jg j t

t V

L V

p sp

p p

CDt kg k kg k CDt kg k

p p p

L V L V L V

spsp psp

with D diam and from we obtain A when

t C diam kg k and for all smaller t

L V

p p p p

A C diam t kg k

This completes the pro of of inequality Inequality follows from Theorem

To prove inequality we take L norm on the b oth sides of inequalities

and and then we apply Maximal Theorem we use the fact that qp

Case p s Notice rst that

exp t

Secondly and the Holder inequality give for each integer k

k s

g k k C diam kg k kJ

L V

By keeping go o d track of the constants app earing in the pro of of one can check

that C C sbC k The desired inequality follows by summing over k We leave

the details to the reader as we prove a b etter estimate in the next section under slightly

stronger assumptions

Case p s The lower b ound on gives

i p p

B x jg j d J g x

B x

diam

isp sp p

diam kg k C

L V

diam

C diam kg k

L V

The pro of of Theorem is complete

Proof of Theorem All the inequalities but and follow directly from

Theorem and Theorem The Holder continuity estimate follows from

and the lower bound If p then inequality is trivial as it is weaker than

the pPoincare inequality If p it follows from Theorem and from the rst

inequality in Theorem The pro of is complete

Trudinger inequality

n n

When X R and u belongs to the Sob olev class W for a ball one has the

following Trudinger inequality

C ju u j

dx C exp

jjrujj

Here C and C dep end only on the dimension n As in case of the Poincare inequality

the exact value of u is not crucial In fact it is easy to see that we may replace it by

the average of u over some xed ball B In the previous section weobserved that

an sPoincare inequalitywith s not exceeding the lower order of the doubling measure

results in exp onential integrability We do not know if one could get an analog of the

Trudinger inequalityinsuch a general setting but we doubt it

In this section weverify an analog of the Trudinger inequality for connected doubling

spaces Thus the only assumption we need to add is that X b e connected For related

results see Buckley and OShea and MacManus and Perez

Theorem Assume that X is a connected doubling space and that the measure

satises condition with s Suppose that the pair u g satises an sPoincare

inequality Then there are constants C and C such that

C B ju u j

dx C exp

diam B kg k

L B

for any bal l B X

Remarks It is easy to see that as the space is connected the condition cannot

hold with s Weleave the details to the reader The argumentemployed in the

pro of actually shows that the inequality holds with B replaced by B

For the pro of of this theorem we need a chain condition a version of which will also

be used later on

We say that X satises a chain condition if for every there is a constant M

such that for each x X and all r R diam X there is a sequence of balls

B B B B for some integer k with

B X n B x R and B B x r

M diam B dist x B M diam B for i k

i i i

there is a ball R B B such that B B MR for i k

i i i i i i

no p oint of X b elongs to more than M balls B

The sequence fB g will b e called a chain associated with x rR

The existence of a doubling measure on X do es not guarantee a chain condition In

fact such a space can be badly disconnected whereas a space with a chain condition

cannot have large gaps

Let us show that each connected doubling space satises a chain condition Fix

j j j

Write A xB x n B x for r R As is doubling we can cover

eachannulus A xbyatmostN balls of radii equal to with N indep endentofx j

Naturally N dep ends on and the smaller the the larger the number N Consider

the collection of all these balls when r R When is suciently small

dep ending only on the balls B with B corresp onding to A x and B with B

corresp onding to A xdonotintersect provided ji j j The balls B corresp onding

to the annuli A x together with B x r X n B x R form on op en cover of X

As X is connected and contains a point inside B x r and another p oint outside

B x R we can pick a chain of these balls B that joins B x r to X n B x R

The required chain is then obtained as the collection of the balls B from the balls B

dierent from B x r in this chain

Lemma Assume that X satises a chain condition and suppose that a pair u g

satises an sPoincareinequality for al l bal ls in X Then the fol lowing holds for almost

every x Let Rdiam X Thereis r and a chain B corresponding to x rR

with such that

g d jux u j C r

B i

Proof As is xed conditions and of the denition of the chain and the Leb esgue

dierentiation theorem see Theorem guarantee that u ux for almost all

x when r tends to zero here k k For such a point we have for appropriate r and

corresp onding k

jux u j ju u j

B B B

i i

u j ju u j ju

R B R B

i i i i

Z Z

ju u j d ju u j d

B B

i i

R R

i i

C ju u j d

C r g d

The pro of is complete

Proof of Theorem By the discussion preceding the previous lemma we know

that X satises a chain condition Thus we may assume that the p ointwise inequality

holds for a given p oint x Write r for the radius of the xed ball B Wemay assume

that diam B rC and that B B B B for each i

i i

Fix qmaxfs ss g For q we have

g d jux u j C r

B i

Z Z

sq q

q s s s

r B r C g d g d

B B

i i

As s s q s q we can use Holders inequality to estimate

X X

q s

q s s

jux u jC r B M g x r kg k

B i B

L B

i i

s s

g d by CM g x and used the bounded overlap of the balls here we replaced

s s

B to replace the sum of the integrals of g over these balls by the integral of g over

To estimate the second term in the pro duct we sum over the balls B that corre

sp ond to an annulus A let us write I for the set of indices i corresp onding to A

j ij j

By the construction of the chain we know that we have at most N balls for each xed

j Moreover the radii of balls corresp onding to dierent A form a geometric sequence

and hence

X X X

q q

s s

r M g x M g x r

B B

i i

i j

s q s

CM g x C q r M g x

B B

where C is indep endentofq In the last inequalityweemployed the fact that q

For the rst term we use the lower bound on B and argue as ab ove

X X

sq ss

q s sq q s ss

r r B C r B

i i

q s

r B Cqs q

where C is an absolute constant

If we let sq then q s q q s sq q as q ss Hence

q s q ss s

jux u j C kg k B q r M g x

B B

L B

here C is an absolute constant

We pro ceed to estimate the integrals of ju u j By the triangle inequality

ju u j ju u j ju u j

B B B B

By controlling the second term bythePoincare inequality and using the ab ove p ointwise

estimate for the rst term we arriveat

Z Z

q q sqs qs q s

ju u j d C q B kg k r M g d

B B

L B

B B

q q qs

C r B kg k

L B

By the Maximal Theorem see Theorem and Theorem in the app endix

Z Z

s s

M g d C B g d

B B

and hence we conclude that

Z Z

s q q sqs s

r ju u j d C q g d

B B

where C do es not dep end on q Notice that this estimate holds as well for q

maxfs ss g by Theorem

Now

kss

tjux u j

exp tjux u j

Integrating over B and using the ab ove estimate we obtain

kss

s k

tr B kg k k Ck exp tjux u j d

L B

This series converges when tr B kg k C where C dep ends only on

L B

C s and the claim follows

A version of the Sob olev emb edding theorem on

spheres

In order to state our version of the Sob olev emb edding theorem on spheres we rst

have to deal with the problem that u is only dened almost everywhere To take care

of this matter we dene ux everywhere by the formula

ux lim sup uz dz

B xr

As almost every point is a Leb esgue point we have only mo died u in a set of

measure zero This redenition of u essentially corresp onds to picking a representative

of u with nice continuity prop erties for related results see Ha jlasz and Kinnunen

and Kinnunen and Martio

We again assume that X is a doubling space and

B x r C B

whenever B x r B B x r Recall that such an estimate follows from the

doubling condition

Theorem Suppose that the pair u g satises a pPoincare inequality and that

p s Then the restriction of u to the set fx dx x r g is uniformly Holder

continuous with exponent s p for almost every r r In particular there

is a constant C and a radius r r r such that

p p sp

g d jux uy j C dx y r

whenever dx x dy x r The constant C only depends on p s C C C

P b d

In the case of Carnot groups a related result has b een indep endently obtained by

Vo dopyanov

The usual Sob olev emb edding theorem on spheres cf Lemma is based

on showing that the trace of a Sob olev function b elongs to a Sob olev class on almost all

spheres One then uses the Sob olev emb edding on the sphere that is lower dimensional

than the ball In our situation a sphere can be very wild and this approach cannot be

used We prove the ab ove result using a maximal function argument

The reader may wonder why the integration is taken over all of B and not over

an annulus The reason for this is that p oints on the sphere cannot necessarily b e joined

inside an annulus centered at x For a trivial example let X be the real axis If we

assume that X has reasonable connectivity prop erties we obtain a stronger conclusion

Theorem Suppose that the pair u g satises a pPoincare inequality and that

p s Assume that any pair of points in B n B can be joined by a continuum

F in CB n C B with diam F Cdx y Then there is a constant C and a radius

r r r such that

sp p p

jux uy j C dx y r g d

C B nC B

whenever dx x dy x r The constant C only depends on p s C C C C

P b d

By combining Prop osition and Theorem we obtain the following corollary recall

that a pPoincare inequality guarantees a q Poincare inequalitywhen qp

Corollary Suppose that X is weakly local ly quasiconvex and that C r

B x r Cr with s for each x and al l r Let s p s Assume that

for each pair u g of a function and its upper gradient we have a pPoincare inequality

Then there is a constant C and a radius r r r such that

p p sp

g d jux uy j C dx y r

C B nC B

whenever dx x dy x r The constant C only depends on p s C C

In the preceding corollary we assumed that s and that p s Both of these

assumptions are necessary Indeed the Poincare inequality holds for the real axis

but one needs to integrate over balls instead of annuli The union of the two closed

cones in IR with a common vertex of Example supp orts a pPoincare inequality for

all p n and B x r r for each x and all r One again needs to integrate over

balls instead of annuli

Before pro ceeding with the pro ofs of Theorems and let us discuss one more

application We say that u is monotone if

sup jux uy j supfjuy uw j dx y dx w r g

ywB xr

for each ball B x r Supp ose that u is monotone u has an upp er gradient in L and

the assumptions of the previous corollary hold Then u is continuous and

C M

kg k jux uy j C log

sB xC M

dx y

for all x y with dx y M This estimate is commonly used in the Euclidean setting

We leave it to the reader to deduce this conclusion from the ab ove corollary

Proof of Theorem Fix x y X and Set B B x dx y and

i i

dene B B x dx y when i and B B y dx y when i

i i

Then using the Poincare inequality and the triangle inequality as in the pro of of

Theorem we have

jux uy j C ju u j d

C r g

C r r g

i i

M g xM g y Cdx y

dxy p dxy p

where

g d M g x sup r

B xr

Observe that according to the ab ove inequality holds everywhere cf

Write G fx B M g x tg Then

t r p

jux uy j Ctdx y

for all x y G

By the covering Lemma and the lower bound on B x r

p sp p s

g d H B n G Ct r

Recall that the Hausdor content H E is dened as the inmum of r

where the inmum is taken over the set of all countable coverings of the set E by balls

with radii r

Dene v B r by the formula v x dx x Then v is Lipschitz continu

ous with constant and hence for each set E B

sp sp

H v E H E

Let s p Then

p p s

g d H v B n G Ct r

This implies that the length of the set v B n G r goestoas t go es to Now

the theorem follows from the observation that for r r n v B n G the sphere

fx dx x r g is contained in G and hence inequality applies The pro of is

complete

Proof of Theorem Join the points x y by a continuum F in A CB n B C

with diam F Cdx y Let r C dx y and consider the collection of

all balls B w r with w A B x Cdx y As is doubling we nd a cover of

A B x Cdx y consisting of k of these balls with k dep ending only on C C

Pick those balls from this cover that intersect F and order them into a chain That

is denoting the balls by V V V for i l and x V y V

i i i l

assuming that we have l balls The claim of Theorem follows rep eating the pro of

of Theorem with the following mo dication we dene B V for i l

i i

i il

B B x C dx y for i and B B y C dx y for

i i

i l It is helpful to notice here that the balls B for i l have uniformly

bounded overlap as l k

RellichKondrachov

The classical RellichKondrachov emb edding theorem states that given a b ounded

domain IR with smo oth b oundary the Sob olev space W p is

compactly emb edded into L where q is any nite exp onent when p n and

any exp onent strictly less that npn p when p n Of course here the Sob olev

space W is dened in the classical way

In the case of Sob olev spaces asso ciated with vector elds some compact emb edding

theorems have been obtained by Danielli Franchi Serapioni and Serra Cassano

Garofalo and Lanconelli Garofalo and Nhieu Lu Manfredini

Rothschild and Stein

In this section we extend the RellichKondrachov theorem to the setting of metric

spaces As we will see in Section Sob olev inequalities for vector elds are sp ecial

cases of Sob olev inequalities on metric spaces Hence our result covers many of the

ab ove results It extends also an earlier result of Ha jlasz and Koskela from the

Euclidean setting In the case of Sob olev spaces on metric spaces intro duced byHajlasz

a related compactness theorem has b een proved indep endently by Kalama jska

Let be a Borel measure on X doubling on As usual X denotes an

op en subset of a metric space In order to prove the compactness theorem for Sob olev

functions on we need to assume that a kind of emb edding theorem holds on Thus

until the end of the section we make the following assumption

The op en set X satises and there exist exp onents p and q

such that every pair u g which satises a pPoincare inequality in with given

constants C satises also the global Sob olev inequality

Z Z Z

q p

juj d C juj d g d

Observe that follows form the Sob olevPoincare inequality

Theorem Let X p and q be as above Let fu g g be a sequence of

i i

pairs al l of which satisfy the pPoincareinequality in with given constants C

If the sequence ku k kg k is bounded then fu g contains a subsequence that

i i i

w q

converges in L for any wq to some u L

Proof Let fu g g be a sequence satisfying the assumptions of the theorem Since the

i i

sequence fu g is b ounded in L we can select a subsequence still denoted by fu g

i i

q q

weakly convergent in L to some u L It remains to prove that this sequence

converges to u in the norm of L for every wq

Lemma Let Y be a set equipped with a nite measure Assume that fv g

q q

L Y q is a bounded sequence If v converges in measure to v L Y

then v converges to v in the norm of L Y for every wq

The lemma is a variant of Prop osition We p ostp one the pro of of the lemma for a

moment and we show how to use it to complete the pro of of the theorem According

to the lemma it remains to prove that the functions u converge to u in measure

Assume that otherwise the pro of is even simpler For t set fx

dist x tg Fix and t For ht recall that app ears in

and x we set

Z Z

u x ud and u x u d

h ih i

B xh B xh

We have

fx ju uj g fx ju u j g

t i t i ih

fx ju u j g

t ih h

fx ju u j g

t h

A B C

ih ih h

Note rst that

p p

ju x u xj Ch M g x Ch M g x

i ih h

i i

by for almost every x Thus the maximal theorem see Theorem gives

p p

C g d A M g C

i i

This convergence is uniform with resp ect to i as the sequence kg k is b ounded

i L

It follows from the denition of the weak convergence in L that for every x

u x u x as i so B as i Finally C by the Leb esgue

ih h ih h

dierentiation theorem see Theorem Now it easily follows that u u in

measure Thus the pro of is completed provided we prove the lemma

Proof of the lemma Fix w q It suces to prove that every subsequence

of fv g contains a subsequence convergent to v in L Y In what follows all the

subsequences of fv g will b e simply denoted by fv g Take an arbitrary subsequence of

i i

fv g The convergence in measure implies that this subsequence contains a subsequence

whichisconvergent almost everywhere Then by Egorovs theorem for any there

exists a measurable set E Y with the prop erty that Y n E and v converges

to v uniformly on E Hence

Z Z

w w q q

jv v j d Y n E jv v j d

k j k j

Y Y nE

jv v j d

k j

w q w

C jv v j d

k j

w q

which gives lim sup kv v k C Since was arbitrary the

k j

L Y

subsequence fv g is a Cauchy sequence in L Y and hence the lemma follows This

completes also the proofofthe theorem

Below we state another version of the compactness theorem The pro of follows by

some obvious mo dications to the ab ove pro of

Theorem Let X be a doubling space and let s be the lower decay order of the

measure from Suppose that al l the pairs u g satisfy a pPoincare inequality in

i i

X with xedconstants C Fix a bal l B and assume that the sequence ku k

P i

L B

kg k is bounded Then thereisasubsequenceoffu g that converges in L B for

i L B i

each qpss p when ps and for each q when p s

Notice that this theorem gives compactness in the entire space provided the space has

nite diameter

We would liketothank Agnieszka Kalama jska for an argument that simplied our

original pro of of the compactness theorem

Sob olev classes in John domains

In the pPoincare inequality we have allowed g to be integrated over a larger ball

than u is integrated over One cannot in general reduce the radii of the balls on the

right hand side To see this consider the following example let

and u on u on g on and g const is

very large on The details are left to the reader

Hence in the Sob olev typ e inequalities like Theorem or Theorem wehaveto

integrate g over a larger ball as well

Weshow in this section that one can use balls of the same size provided the geometry

of balls is suciently nice This leads us to dene John domains

The rst subsection is devoted to study of the geometry of John domains and in

the second subsection we study Sob olev inequalities in John domains

John domains

When dealing with Sob olev typ e inequalities in domains in IR one usually assumes

that the domain is nice in the sense that its b oundary is lo cally a graph of a Lipschitz

function This notion of b eing nice is not appropriate for the setting of metric spaces

and so one has to dene a nice domain using only its interior prop erties This leads

to John domains

Denition A b ounded op en subset of a metric space is called a John domain

provided it satises the following twisted cone condition There exist a distinguished

point x and a constant C such that for every x there is a curve

l parametrized by the arclength and such that x l x and

dist t Ct

The length l dep ends on x

Notice that every rectiable curve in a metric space can be parametrized by ar

clength see Busemann or Vaisala

John domains in IR were intro duced by Martio and Sarvas They are named

after F John who considered similar domains in

The class of John domains in IR is much larger than the class of domains with the

interior cone condition In general the Hausdor dimension of the b oundary of a John

domain can be strictly larger than n

The ab ove denition of John domain is still not appropriate for many metric spaces

as p oints in an arcwise connected metric space may not b e joinable by rectiable curves

For example if IR is the von Ko ch snowake curve and is a nontrivial

sub curve then is not a John domain However we would like to include at least

some of the metric spaces that lack rectiable curves in the class of John domains see

Example The following denition seems to give a prop er generalization of John

domains

Denition A b ounded op en subset of a metric space X d is called a weak John

domain provided there is a distinguished p oint x and a constant C such

that for every x there exists a curve such that x x

and for every t

dist t C dx t

c c

We call such a curve a weak John curve If then we set dist t

and hence is always satised

Notice that this denition can b e also used in the setting of quasimetric spaces ie

when the triangle inequality is replaced by x y K x z z y K In

general it can happ en that even a very nice metric space do es not contain nontrivial

rectiable curves With the metric x y jx y j on the real axis any interval is

of innite length

Example If f S S is a quasiconformal mapping and S is any smo oth

Jordan curve then any connected part of f is a weak John domain This

includes the class of Jordan curves S such that b oth comp onents of S n are

John domains see Nakki and Vaisala

There are also many other fractal sets whose nice subsets are weak John domains

while they cannot be John domains b ecause of the lack of rectiable curves

Example Let X b e a b ounded arcwise connected metric space If wetakeX

then is weak John domain since is satised for any curve joining x and x

The reader may nd the preceding example somewhat articial Let us briey

clarify the issue Our aim is to deduce a pPoincare inequalityforwhenever such an

inequality holds for all balls in that is B x r If we are given an underlying

space X then we consider the balls of the space X that are contained in Otherwise

the collection of the balls can b e fairly large For example let b e a b ounded domain

in IR so that equipp ed with the restrictions of the Euclidean distance and volume

is a doubling space If we neglect IR and consider as our entire doubling space we

shall obtain a Poincare inequality for provided we assume aPoincare inequality for

the pair u g for all balls of These balls consist of the intersections of all Euclidean

balls centered in with see Corollary

It is known that in the Euclidean case X IR the class of weak John domains

coincides with the class of John domains see Vaisala Theorem In a metric

space this is no longer true Clearly every John domain is a weak John domain but the

converse implication may fail However we generalize the result byproving that under

a mild additional condition on the space X the ab ove two denitions are equivalent

see Prop osition

The crucial prop erty of John domains for us is that they satisfy a chain condition

that is essential in order to eectively use the Poincare inequality on the balls contained

in the domain

Let us slightly mo dify the chain condition we employed in connection with the

Trudinger inequality

Denition We say that satises a C M condition where M if there is

a central ball B such that to every x and every r there is a sequence

of balls B B B B B is xed and k may dep end on x r with the following

prop erties

B for i k and B B x r

i k

M r dist x B Mr for i

i i i

there is a ball R B B such that B B MR for i

i i i i i i

no p oint of b elongs to more than M balls B

Avariant of the ab ovechain condition was employed in Ha jlasz and Koskela

Theorem Assume that X is a metric space which is doubling on X If

X is a weak John domain then satises the C M condition for any

with some M depending on C and doubling constant only

Proof Assume rst that X Let B B x dist x Assume that x is

far away from B Say x n B Let be a weak John curve First we dene a

B sequence of balls B as follows The centers x of all balls B lie on Let B

i i

Then we trace along starting from the Assume that we have already dened B

center x towards xuntil weleave B for the last time We let this p oint b e the center

x of B and we dene

r dx x

Now we dene B B Prop erties and are evidently satised provided we

cho ose k large enough Prop erty follows from the fact that consecutive balls have

comparable radii and x lies on the b oundary of B ball R is centered at x and

i i i

of radius equal to minimum of r r For prop erty assume that y B B

i i

It follows from the construction that the radii of the balls B j l B

i i

and the distances b etween centers of the balls are all comparable to dx y Indeed the

radii are comparable and the distance of the centers times are no less than the radii

are pairwise disjoint Thus there exists a constant t such that tB tB tB

i i i

and all these balls are contained in a ball centered at y and having radius comparable

to dx y so the doubling condition implies the upp er bound for l

The case when x B is similar If x n B we dene B B and the rest of

the argumentgoesasabove Otherwise we consider the union of twocurves the weak

John curveforx and the weak John curve for some p oint y with dy x dist x

This curve traced rst from x to x and then from x to y is easily seen to b e a John

curve with a new distinguished point y and a new John constant only dep ending on

C One can then dene desired chain by rst replacing the role of x in the ab ove

argument by y and then adding a chain joining y to x We leave the details to the

reader

Supp ose nally that X Then X is b ounded We x x and dene

B B x diam X The rest of the argument is the same as in the case X

except that in we replace C by The pro of is complete

Lemma Let X d be an arcwise connected metric space such that bounded and

closed sets in X are compact Assume that the metric d has the property that for

every two points a b X the distance da b is equal to the inmum of the lengths

of curves that join a and b Then there exists a shortest curve from a to b with

da bda z dz b for every z

Remark Observe that the compactness of the sets that are b ounded and closed is a

stronger assumption than the space being lo cally compact The two conditions are

equivalent if the space is complete

This lemma is due to Busemann page cf page The idea of the

pro of is the following Let f g be a sequence of rectiable curves joining a with b

and such that the length of converges to da b Parametrize each by arclength

k k

Scaling the arclength parametrizations wemay assume that all curves are dened on the

interval Now it easily follows that the family f g is equicontinuous b ecause of

the go o d parametrization By a standard diagonal metho d we can nd a subsequence

of f g which converges on a dense subset of The equicontinuity implies the

uniform convergence on the whole interval It is easy to prove that the length of the

limiting curve is da b For a detailed pro of see page

Corollary Let the metric space X d satisfy the assumptions of the above lemma

Then each bal l B X is a John domain with a universal constant C

This corollary shows that balls in a CarnotCaratheo dory metric see Section are

John domains

The chain condition is closely connected to the concept of a John domain as the

following prop osition shows Analogs of this result can be found in Buckley Koskela

and Lu and in Garofalo and Nhieu where the authors employa Bomanchain

condition that is dierent from ours

Prop osition Let X be a metric space which is doubling on X Assume that

has the fol lowing local connectivity property there exists a constant such that for

every bal l B with B every two points x y B can be connected by a rectiable

curve contained in B and of length less than or equal to dx y Then the fol lowing

three conditions are equivalent

is a John domain

is a weak John domain

is a C M domain for each and for some M

Proof The implications hold without any lo cal connectivity assumptions

on the rst implication is immediate from the denitions and the second one follows

be an from Theorem We prove the implication Fix x and let fB g

c c

asso ciated chain for and r dx We dene B B x dx

If the radii of the balls B were to increase geometrically when i decreases we would

obtain the John curve simply by joining the centers of the balls in our chain by curves

obtained from the lo cal connectivity condition However this need not be the case

This dicultycan be rectied as follows

If the entire chain is contained in B x C dx the b ounded overlap condition

for the chain and the doubling prop erty imply that the length of the sub chain ie the

numb er of balls joining the b oundary of B to x do es not exceed a uniform constant

that dep ends on the doubling constant the constants in the chain condition and on

C The ab ove connectthedots argument applies in this case Otherwise we consider

the sub chain joining B to Bx C dx Again the length of this sub chain is

bounded by a uniform constant and the radii are bounded from below by a multiple

c c

of dx Pick a point y with dy x Cdx that is contained in one of the

c c

balls of the sub chain If the constant C is suciently large then dy dx

Consider a chain joining y to x Wenow rep eat the ab ove argument for the sub chain

joining y to Bx C dx By continuing inductively we obtain a new chain with

the appropriate geometric b ehavior We leave it to the reader to provide the details

Sob olev typ e inequalities

In the main theorem of this section Theorem we show how the claims of Theo

rem and Theorem extend to the setting of John domains

The study of the Sob olev typ e inequalities in John domains in IR originated in the

pap ers of Boman Bo jarski Goldshtein and Reshetnyak Hurri

Iwaniec and Nolder Kohn and Martio It was then generalized to the

CarnotCaratheo dory spaces by Jerison and then to more general situations by

Franchi Gutierrez and Wheeden Garofalo and Nhieu Ha jlasz and Koskela

Lu Other related references include Buckley and Koskela

BuckleyKoskela and Lu Chen and Li Chua HurriSyrjanen

Ha jlasz and Koskela Franchi Maheux and SaloCoste Salo

Coste Smith and Stegenga

Buckley and Koskela showed that the class of John domains is nearly the

largest one for which one can prove the Sob olevPoincareemb edding theorem

Theorem Let X be a metric space equipped with a measure which is doubling on

a weak John domain X Assume that the measure satises the condition

B x r C

diam

whenever x and r diam If the pair u g satises a pPoincare inequality

p in then al l the claims of Theorem hold with B and B replaced by

For example we get that if p s and the pair u g has the truncation property

then

Z Z

p p

C diam inf ju cj d g d

cIR

where p sps p

If in addition the space is connected and p s then the Trudinger inequality

holds in ie

C jj ju u j

exp d C

diam kg k

The constants C C C depend on p s C C C and C only

P d b J

Remarks As follows from the pro of the ab ove theorem actually holds for any

op en set that satises the C M condition We have stated explicitly only a

generalization of one part of Theorem inequality It is left to the reader to

formulate generalizations of the other cases Observe that ju u j is replaced by

ju cj and inmum over c IR is taken This is necessary if p as then we cannot

apply inequality

Proof By Theorem the domain satises the chain condition for any given

Thus we obtain inequality with balls B as in the chain condition in

particular with B B The pro ofs of Theorem and Theorem give the claim

Corollary Let X be a doubling space satisfying the assumptions of Lemma

Suppose that the measure satises condition Then al l the claims of Theorem

hold with the integrals of g taken over B instead of B If in addition the space is

connected and s then the Trudinger inequality holds with the integral of g

taken over B

Remark This corollary applies to the CarnotCaratheo dory spaces see Prop osi

tion

Proof By Corollary every ball is a John domain with a universal constant C

and hence we may apply Theorem The pro of is complete

We have already mentioned that any b ounded arcwise connected set X is a

weak John domain To illustrate this issue we state the following sp ecial case of the

ab ove results

Corollary Let IR be an arbitrary bounded domain Assume that jB x r

n p

j Cr whenever x and r diam Assume that u W p n

satises

Z Z

ju u j dx Cr jruj dx

B B

whenever B B x r x and r diam Then the global Sobolev inequality

Z Z

p p

ju u j dx C diam jruj dx

holds where p npn p

Proof Take X The condition jB x r j Cr means that the space X

equipp ed with the Leb esgue measure and the Euclidean metric is doubling Since

X we conclude that is a weak John domain and hence the claim follows from

Theorem The pro of is complete

As the last application of the chain metho d we improve the so called representation

formula The result b elow is a generalization and a simplication of earlier results

due to Cap ogna Danielli and Garofalo Franchi Lu and Wheeden and

Franchi and Wheeden

Theorem Assume that X is a metric space which is doubling on a weak John

domain X If the pair u g satises a pPoincare inequality p in then

for almost every x we have the inequality

p p

g y dx y

jux u jC dy

B x dx y

A x

j j c

where A x B x n B x and B B x dist x x is a

xed bal l

In particular when p we get

g y dx y

jux u j C dy

B x dx y

Proof of Theorem By Theorem the domain satises the C M condition

with Then we have

jux u j ju u j

B B B

i i

C r g d

Each ball B is covered by a nite numb er say no more than l of the annuli A x

i j

Hence if B A x we get

i j

Z Z

j l

p p

g y dx y

dy g d C r

B x dx y

B A x

j l

Now observe that the doubling condition and the b ounded overlapping of the balls B

implies that the number of balls B with B A x is bounded by a constant

i i j

not dep ending on j This easily implies the claim

There are several related results when p For the Euclidean case see Goldshtein

and Reshetnyak Martio and Ha jlasz and Koskela for the Carnot

Caratheo dory case see Franchi Lu and Wheeden Cap ogna Danielli and Garofalo

and for the case of doubling spaces see Franchi Lu and Wheeden Franchi

and Wheeden

Poincare inequality examples

The purp ose of this section is to illustrate the results obtained up to now in the pap er

we collect basic examples of pairs that satisfy pPoincare inequalities

We will pay particular attention to the validity of the truncation prop erty Recall

that this prop erty is used to prove the Sob olev emb edding in the b orderline case with

the sharp exp onent

Two classes of examples CarnotCaratheo dory spaces and graphs require a longer

presentation and so we discuss them in Sections and

Riemannian manifolds

The pair u jruj where u Lip IR satises the Poincare inequality and hence all

the pPoincare inequalities for p Obviously the pair u jruj also has the

truncation prop erty

This result extends to those Riemannian manifolds whose Ricci curvature is b ounded

from b elow Let M b e a complete Riemannian manifold of dimension n and let g denote

the Riemannian metric tensor Denote the canonical measure on M by Assume that

the Ricci curvature is b ounded from b elowie Ric Kg for some K Then the

BishopGromov comparison theorem implies that

B x r exp n K r B x r

see Cheeger Gromovand Taylor Moreover Busers inequality implies that

Z Z

Krr jruj d ju u j d C n exp

B B

This shows that for any R b oth the doubling prop erty and the Poincare inequality

hold on all balls with radii less than R If we assume that the Ricci curvature is

nonnegative ie K then we can take R Obviously the pair of a Lipschitz

function and the length of its gradient has the truncation prop erty in this setting as

well

Thus the results of our pap er imply that in the ab ove setting the Sob olevPoincare

inequality holds see Maheux and SaloCoste and also Ha jlasz and Koskela

An excellentintro duction to the Buser inequality and the comparison theorems can

be found in Chavels b o ok

For related and earlier works on Poincare and Sob olev inequalities on manifolds

with a bound on the Ricci curvature see Chen and Li Gallot Kusuoka and

Stro o ck Li and Scho en Li and Yau SaloCoste

Upp er gradients

Let X d be a metric space with a Borel measure not necessarily doubling

Denition We say that aBorelfunction g isan upper gradient on of

another Borel function u IR if for every Lipschitz curve a b wehave

ju b u aj g t dt

Note that g is an upp er gradientofany Borel function u

Denition We say that the space supports a pPoincare inequality inequality on

p if every pair u g of acontinuous function u and its upp er gradient g on

satises the pPoincare inequality in with some xed constants C

If wesay that the space supp orts a pPoincare inequalitythenwe mean that ab ove

Since every rectiable curve admits an arclength parametrization that makes the

curve Lipschitz the class of Lipschitz curves coincides with the class of rectiable

curves mo dulo a parameter change

It is necessary to assume that the function g is dened everywhere as we require

the condition for al l rectiable curves We refer the reader to Busemann or

Vaisala Chapter for more information on integration over rectiable curves

The notions of an upp er gradient and a space supp orting a pPoincare inequality

were intro duced by Heinonen and Koskela and then applied and further devel

op ed by Bourdon and Pa jot Cheeger Franchi Ha jlasz and Koskela

Heinonen and Koskela Kallunki and Shanmugalingam Koskela and Mac

Manus Laakso Semmes Tyson and Shanmugalingam

Notice that ab ove we required the pPoincare inequality for continuous functions

and their upp er gradients If X is suciently nice say quasiconvex and each closed ball

in X is compact then the pPoincare inequality follows for any measurable function and

its upp er gradient In fact it would in such a setting suce to assume the pPoincare

inequality for Lipschitz functions and their upp er gradients For this see

Prop osition If u is a Lipschitz function on a Riemannian manifold M then any

measurable function g such that g jruj everywhereisanupper gradient of u On the

p p p

other hand if g L M is an upper gradient of u L M then u W M and

g jruj almost everywhere

Proof The rst part of the prop osition is immediate the second one follows from

the ACL characterization of the Sob olev space see for example Ziemer Theo

rem

Remark It is not true in general that any upp er gradient g of u C M

satises g jruj ae unless we assume that g is lo cally integrable As an example

take ux x on Let E be a Cantor set with p ositive length and set

g x if x E g x if x E One can then improve this example and even

obtain g everywhere

Prop osition If u is a local ly Lipschitz function dened on an open subset of a

metric space X then the function jr ujx lim sup jux uy jdx y is an

y x

upper gradient of u

Remark The prop osition is no longer true if we only assume that u is continuous

Indeed if u is the familiar nondecreasing continuous function u such

that u u and u is constant on connected comp onents of the complement

of the Cantor set then jr uxj ae in

Proof of the proposition Let a b be Lipschitz The function u is

Lipschitz and hence dierentiable ae It easily follows that ju tjjr u tj

whenever u is dierentiable at t Hence

Z Z

b b

ju b u aj ju tj dt jr u tj dt

a a

The pro of is complete

Theorem Assume that the space X supports a pPoincare inequality on Then

any pair u g of a continuous function and its upper gradient on has the truncation

property

Proof Let g be an upp er gradient of a continuous function u We have to prove a

family of pPoincare inequalities for all the pairs v g where v u b

ft v t g

see the denition of the truncation prop erty Since g is an upp er gradient of each of

the functions v we can assume that u v Thus it remains to prove the inequality

Z Z

t p t

ju g d u j d Cr

ft ut g

t t

B B

The following lemma is due to Semmes Lemma C For readers convenience

we recall the pro of

Lemma Let g be an upper gradient of a continuous function u Let t

t and let V be an arbitrary open set such that ft u t g V Then g is

an upper gradient of u

Proof Let be a curve as in the denition of the upp er gradient We have to prove

the analog of for u and g If either is contained in V or is contained in

X nft u t g the claim is very easy In the general case the curve splits into a

nite numb er of parts each of whichiscontained in V or in X nft u t g and the

lemma follows by applying the preceding sp ecial cases to those pieces of

Now we can complete the pro of of the theorem Take t s s t Then

fs u s gV where V ft ut g Applying the pPoincare inequality to

the pair u g and passing to the limit as s t s t we obtain the desired

inequality This completes the pro of

Theorem is interesting provided we can nd suciently many examples of non

smo oth metric spaces that supp ort a pPoincare inequality The rest of Section is

devoted to the discussion of such examples

Top ological manifolds

Denition A metric space X is called Qregular Q if it is complete metric space

Q Q

and there is a measure on X so that C r B x r C r whenever x X and

r diam X

It is well known that one can replace in the ab ove denition bytheQdimensional

Hausdor measure see for example Lemma C

Semmes proved a pPoincare inequality on a large class of Qregular metric

spaces including some top ological manifolds

The following result is a direct consequence of a more general result of Semmes

Theorem Let X be a connected Qregular metric space that is also orientable

topological Qdimensional manifold Q integer Assume that X satises the local

linear contractibility condition there is C so that for each x X and R

C diam X the bal l B x R can be contracted to a point inside B x C R Then the

space supports a Poincare inequality

For related inequalities also see David and Semmes and Semmes

Gluing and related constructions

Heinonen and Koskela Theorem proved that gluing two spaces that supp ort

a pPoincare inequality along a suciently large common part results in a new space

that also supp orts a pPoincare inequality For example one can glue two copies of the

unit ball of IR along the usual Cantor set and the resulting doubling space supp orts

log

This pro cedure allows one to build plethora a pPoincare inequality for all p

log

of examples Hanson and Heinonen used this typ e of a construction recently to

build a space that supp orts the Poincare inequality but has no manifold p oints

Laakso constructed recently for each Q a Qregular metric space that

supp orts the Poincare inequality Notice that here Q need not be an integer These

spaces do not admit biLipschitz imb eddings into any Euclidean space They are ob

tained as quotients by nite to one maps of pro ducts of intervals with Cantor sets

The rst authors to nd noninteger dimensional Qregular spaces that supp ort a

Poincare inequality were Bourdon and Pajot Their examples are b oundaries of

certain hyp erb olic buildings

Further examples

A huge class of examples of spaces that supp ort pPoincare inequalities is contained

in the class of socalled CarnotCaratheo dory spaces that are discussed in Section

This class includes the Carnot groups that have been mentioned ab ove

One can also investigate pPoincare inequalities on graphs see Section Here the

situation is however dierent Since the space is disconnected the notion of an upp er

gradient is absurd Moreover the truncation prop erty do es not hold and it has to be

mo died

There are also many other examples that will not be discussed in the pap er The

main class of such examples is given by Poincare inequalities on Dirichlet spaces

Roughly sp eaking we are given a pair u g satisfying a pPoincare inequality on a dou

bling space and in addition g is related to u in terms of a Dirichlet form see Biroli

and Mosco Jost Sturm Thus

this example ts precisely into the setting of our pap er However the presence of the

Dirichlet form gives additional structure that may lead to results not under the scop e

of our more general setting

The analysis of Dirichlet forms is closely related to the analysis on fractals and

esp ecially with the sp ectral theory of the Laplace op erators see eg Barlow and Bass

Jonsson Kozlov Kigami Kigami and Lapidus

Lapidus Metz and Sturm Mosco As it follows form a recent

work of Jonsson the sp ectral theory of the Laplace op erators on fractals is also

related to the theory of function spaces on fractal subsets of IR develop ed by Jonsson

and Wallin see also Trieb el Some connections with the theory

presented in this pap er seem evident but a b etter understanding of those connections

is still lacking

CarnotCaratheo dory spaces

In this section we give an intro duction to the analysis of vector elds one of the

main areas where the theory of Sob olev spaces on metric spaces is applicable

In the rst subsection we dene the so called CarnotCaratheo dory metric asso ci

ated with a family of vector elds X X X Then in the second subsection

weprove that with resp ect to this metric the gradient jXuj asso ciated with the given

family of vector elds b ecomes the smallest upp er gradient of u We also deal with

Poincare inequalities and Sob olev spaces asso ciated with the given system of vector

elds

The main novelty in our approach is that we develop the analysis on Carnot

Caratheo dory spaces from the point of view of upp er gradients The prime results

of the section are Theorem and Theorem

In the last three subsections we consider Carnot groups and vector elds satisfy

ing Hormanders condition both are examples where pairs u jXuj satisfy such a

Poincare inequality We also discuss some other classes of vector elds that do not

satisfy Hormanders condition but still supp ort Poincare inequalities

CarnotCaratheo dory metric

Let IR be an op en and connected set and let X X X be vector elds

dened in with real lo cally Lipschitz continuous co ecients We identify the X s

with the rst order dierential op erators that act on u Lip by the formula

X uxhX x ruxi j k

j j

We set Xu X uX u and hence

jXuxj jX uxj

With such a family of vector elds one can asso ciate a suitable degenerate elliptic

X X where X is the formal adjoint of X in op erator like for example L

j j

j j j

R R

L ie X uv uX v for all u v C Both the Poincare and Sob olev inequalities

for the pair u jXuj are crucial then for the Harnack inequality for p ositive solutions

to Lu via Mosers iteration Since the Poincare inequality implies the Sob olev

inequality one needs only check the validity of a Poincare inequality Of course this

requires strong restrictions on the class of vector elds

Even if one is concerned with degenerate elliptic equations of the divergence form

LuxdivAxrux

with a symmetric nonnegative semidenite matrix A with smo oth co ecients it is

in general necessary to deal with vector elds that have only Lipschitz co ecients as

they arise in the factorization L X X see Oleinik and Radkevic

For more applications to PDE and references see Section

How do es one proveaPoincare inequality for the pair u jXuj The natural approach

is to bound u by integrals of jXuj along curves and then average the resulting one

dimensional integrals to obtain the desired Poincare inequality

In order to have such b ounds for u in terms of integrals of jXuj one would like to

know that jXuj is an upp er gradientofu Unfortunately this is rarely the case

For example if we have only the single vector eld X x and t t

ux x x in IR then ju u j while jXuj and so jXuj is

not an upp er gradient of u It is not an upp er gradient even up to a constant factor

Roughly sp eaking the problem is caused by the fact that is not spanned bytheX s

There is a brilliant idea that allows one to avoid this problem byintro ducing a new

metric that is describ ed b elow in that makes jXuj an upp er gradientofu onanew

metric space The metric is such that it restricts the class of Lipschitz curves to those

for which is a linear combination of the X s To be more precise it is not always a

metric as it allows the distance to b e innite

We say that an absolutely continuous curve a b is admissible if there

exist measurable functions c t a t b satisfying c t and t

j j

c tX t

j j

Note that if the vector elds are not linearly indep endent at a point then the

co ecients c are not unique

Then we dene the distance x y between x y as the inmum of those T

such that there exists an admissible curve T with x and T y

If there is no admissible curve that joins x and y then we set x y

Note that the space splits into a p ossibly innite family of metric spaces

A where x y A if and only if x and y can be connected by an admissible

i i

curve Obviously A is a metric space and the distance between distinct A s is

i i

innite

If we only have the single vector eld X x in IR then x y jx y j if x

and y lie on a line parallel to the x axis otherwise x y On the other hand

if X x for j n in IR then is the Euclidean metric

j j

The distance function is given many names in the literature We will use the

name CarnotCaratheodory distance A space equipp ed with the CarnotCaratheo dory

distance is called a CarnotCaratheodory space

There are several other equivalent denitions for the CarnotCaratheo dory distance

see eg Jerison and SanchezCalle and Nagel Stein and Wainger The

CarnotCaratheo dory distance can also b e dened for Dirichlet forms see Sturm

It has already b een mentioned that Lipschitz vector elds arise in connection with

degenerate elliptic equations of the divergence form It seems that Feerman and

Phong where the rst to realize that many imp ortant prop erties of the op erator

can be read o from the prop erties of the asso ciated CarnotCaratheo dory metric

Roughly sp eaking they proved that lo cally sub ellipticity of is equivalent to the

estimate x y C jx y j for some

Other connections with degenerate elliptic equations will be discussed later on

in Section We want to emphasize that the scop e of applications of the Carnot

Caratheo dory geometry go es far beyond degenerate elliptic equations and it includes

control theory CR geometry and more recently quasiconformal mappings We refer the

reader to the collection of pap ers for a comprehensiveintro duction to the Carnot

Caratheo dory geometry Other imp ortant references include Franchi Franchi and

Lanconelli Garofalo and Nhieu Gole and Karidi Karidi Liu

and Sussman Nagel Stein and Wainger Pansu SaloCoste

Strichartz Varop oulos SaloCoste and Coulhon to name a few

Lemma Let B x R and let sup jX j M If T

B xR

T RM is an admissible curve with x then T B x R

Proof Assume by contradiction that the image of is not contained in the ball B x R

Then there is the smallest t T such that jx t j R Note that by the

Schwartz inequality j tj jX tj Hence

Z Z

t t

R jx t j t dt jX tj dt MT

which contradicts the assumption T RM The pro of is complete

As as a corollary we obtain the following well known result

Prop osition Let G Then there is a constant C such that

x y C jx y j

for al l x y G

Proof Let x y G and let T x T y be any admissible curve

Fix such that G fx IR dist x G g and set M sup jX j

Obviously B x R G when R minfjx y jg and hence Lemma implies

that T RM minfM M diam G gjx y j This completes the pro of

If x y for all x y then is a true metric called the Carnot

Caratheodory metric Prop osition implies that id jjiscontinuous

However it need not be a homeomorphism as the simple example of the two vector

elds and x in IR shows

x y

In order to avoid such pathological situations it is often assumed in the literature

that

id jj is a homeomorphism

Fortunately is true for a large class of vector elds satisfying the socalled

Hormander condition which includes Carnot groups see the following subsections

and the case of Grushin typ e vector elds like those in Franchi Franchi Gutierrez

and Wheeden and Franchi and Lanconelli

To keep the generality we do not assume unless it is explicitly stated

By a Lipschitz function on we mean Lipschitz continuity with resp ect to the

Euclidean metric in but when wesay that a function is Lipschitz on wemean

Lipschitz with resp ect to the distance The same convention extends to functions

with values in or in Functions that are Lipschitz with resp ect to will b e also

called metric Lipschitz Balls with resp ect to will b e called metric bal ls and denoted

by B

Lemma Every admissible curve T is Lipschitz

Proof Use the Schwartz inequality

Prop osition A mapping T is an admissible curve if and only if

it is Lipschitz ie ba jb aj for al l a b

Proof This implication directly follows from the denition of

Let T be a Lipschitz curve By Lemma it is Lipschitz

with resp ect to the Euclidean metric on and hence it is dierentiable ae Wehaveto

provethat is admissible Let t Tbeany p oint where is dierentiable Since

t t for there exists an admissible curve

t t for any We have

t dt t t t o

By the denition of an admissible curve there are measurable functions c t such that

c t and

k k

X X

t c tX t c t X t X

j j j j j

j j

c tX t at

j j

Note that by Prop osition C j t j t tprovided and are

suciently small Hence jatj jX t X j Ct as the vector elds have

lo cally Lipschitz co ecients Thus we conclude that

t dt t

Z Z

c t dt X t at dt

j j

Selecting suitable sequences and we conclude that t b X t

l l j j

b This completes the pro of

It is well known that any rectiable curve in a metric space admits an arc

length parametrization see or Chapter This also holds for the Carnot

Caratheo dory distance as a CarnotCaratheo dory space splits into a family of metric

spaces such that each rectiable curveisentirely contained in one of these metric spaces

Note also that the arclength parametrization makes the curve Lipschitz and hence

admissible This observation implies the following result

Prop osition The CarnotCaratheodory distance between any two points equals

the inmum of lengths with respect to of curves that join those two points If the

points cannot be connected by a rectiable curve then their distance is innite

Upp er gradients and Sob olev spaces

The following two results generalize Prop osition

Prop osition jXuj is an upper gradient of u C on the space

Proof Let a b beaLipschitz curve By Lemma u is Lipschitz

and hence

Z Z

b b

ju b u aj hru t ti dt jXu tj dt

a a

The inequality follows from the fact that is admissible by Lemma and from the

Schwartz inequality The pro of is complete

Theorem Let g L be an upper gradient of a continuous function u

lo c

on Then the distributional derivatives X u j karelocal ly integrable

and jXuj g ae

The pro of of the theorem is rather complicated and thus we rst make some comments

and give applications and p ostp one the pro of until the end of the subsection

The pro of is particularly easy if u C and the vector elds have C smo oth

co ecients We present it now as it may help to understand the pro of for the general

case

Since u is smo oth we do not havetoworry ab out distributional derivatives and we

simply prove that g jXuj ae

The set of the points where jXuxj is op en Since the desired inequality

holds trivially outside this set we can assume that jXuj everywhere in Let

a xX uxjXuxj and let be anyintegral curveofthevector eld Y a X

j j j j

ie T T t a tX t Obviously is an admissible curve

j j

Thus T T is Lipschitz and hence

ju t u t j g t dt

for any T t t T On the other hand

Z Z

t t

jXu tj dt hru t ti dt ju t u t j

t t

This yields

Z Z

t t

g t dt jXu tj dt

t t

If the vector eld Y were parallel to one of the co ordinate axes then would imply

that g jXuj ae on almost every line parallel to that axis and hence g jXuj ae

in The general case can b e reduced to the case of a vector eld of parallel directions

by the rectication theorem This is obvious if the vector eld is C smo oth as it is

the usual requirement in the rectication theorem see Arnold However the same

argument can b e also used in the general Lipschitz case A construction of the Lipschitz

rectication is provided in the pro of of Theorem lo ok for G

Now we give two applications of the theorem

Note that if u is metric LLipschitz then the constant function L is an upp er

gradient of u However the function u need not be continuous with resp ect to the

Euclidean metric even if the CarnotCaratheo dory distance is a metric This is easily

seen in the previously discussed example of and x in IR Thus in order to

x y

apply Theorem to a metric LLipschitz function we need to assume either that

the function is continuous with resp ect to the Euclidean metric or simply that

holds

The following sp ecial case of Theorem was proved by Franchi Ha jlasz and

Koskela and in a slightly weaker form earlier by Chernikov and Vo dopyanov

Franchi Serapioni and Serra Cassano and Garofalo and Nhieu

Corollary Assume that holds If u is metric LLipschitz then the distri

butional derivatives X u j k are represented by bounded functions and

jXuj L ae

The following version of MeyersSerrins theorem was discovered in its lo cal form by

Friedrichs cf Lemma and later by Chernikov and Vo dopyanov

Lemma Franchi Serapioni and Serra Cassano Prop osition

p and Garofalo and Nhieu Lemma

Since later on we will need estimates from the pro of rather than the statement

alone we recall the pro of following Friedrichs argument

Theorem Let X X X X be a system of vector elds with local ly

Lipschitz coecients in IR and let p If u L and the distri

butional derivative Xu L then there exists a sequence u C such that

p p

ku uk kXu Xuk as k

k k

L L

Proof We will prove that if u has the compact supp ort in then the standard mollier

approximation is a desired approximating sequence The general case follows then by

a partition of unity argument

Let Y x c xx where the functions c are lo cally Lipschitz denote

j j j

n n

one of X s Let x x C B be a standard

mollier kernel For a lo cally integrable function u we have

Y u x c x y x y y dy

x y y dy c x c x y

j j

Yu x

c x c x y y dy ux y ux

j j

Yu xA ux

where the integrals are understo o d in the sense of distributions Note that

c x c x y y CLx ae

j j B

where Lx is the Lipschitz constant of all c s on B x Hence

juy uxj dy jA uxj CLx

B x

If u L has compact supp ort in then it easily follows that kA uk

as Indeed it is obvious if u is continuous and in the general case we can

approximate u by compactly supp orted continuous functions in the L norm If in

p p

addition Yu L then Yu Yu in L and hence byY u Yu

in L The pro of is complete

The following result is a corollary of the ab ove pro of

Prop osition Assume that holds Let u be metric LLipschitz in Then

the standard mol lier approximation converges to u uniformly on compact subsets of

and

jX u xjL jA uxj

where jA uj as uniformly on compact subsets of The above inequality

holds for al l x of distance at least to the boundary

Proof Condition is used to guarantee that u is continuous with resp ect to the

Euclidean metric which in turn together with implies that jA uxj converges

to zero uniformly on compact sets By Corollary jYuj L for any Y

P P

c The desired inequality then results using with the c X with

j j

following choice of the co ecients Fix an arbitrary p oint x If jX u x j

then we are done otherwise we take c X u x jX u x j The pro of is

j j

complete

The theorem below shows that in a certain sense the analysis of vector elds

is determined by the asso ciated CarnotCaratheo dory metric The result is also an

armativeanswer to a question p osed by Bruno Franchi

Theorem Let X and Y be two families of vector elds with local ly Lipschitz

coecients in and such that holds for the induced CarnotCaratheodory metrics

and Then the fol lowing conditions are equivalent

X Y

There exists a constant C such that C C

X Y X

There exists a constant C such that C jXuj jYuj C jXuj for al l

u C

Proof Note that if g is an upp er gradient of u C on then

the equivalence of the metrics implies that Cg is an upp er gradient of u on

This fact Prop osition and Theorem imply that jYuj C jXuj The opp osite

inequality follows by the same argument

Fix x y and let uz x z Let T be an

X Y

arbitrary Lipschitz curve such that x T y

Let u u b e the standard mollier approximation By Prop osition jYu j

is an upp er gradientofu on and hence invoking Prop osition we get

jYu tj dt x y ju T u j

Z Z

T T

C jXu tj dt CT C jA u tj dt CT

Now it follows from the denition of that C The opp osite inequality follows

Y X Y

by the same argument The pro of is complete

Let us come back to the question p osed at the b eginning of the section How do es

one prove a Poincare inequality for the pair u jXuj The natural approach is to

bound the oscillation of u by integrals of jXuj over admissible curves this can be

done as jXuj is an upp er gradientofu Then the Poincare inequality should followby

averaging the resulting line integrals Unfortunately in general this program is very

dicult to handle and it turns out that many additional assumptions on the vector

elds are needed One such a pro of of aPoincare inequalitywillbe presented later on

see Theorem Anyhow if one succeeds in proving a Poincare inequality using

the ab ove idea the resulting inequality holds on metric balls

Thus the Poincare inequalitywe should exp ect is

Z Z

ju u j dx C r jXuj dx

e e

whenever B and u C B Here C p are xed

constants and as usual B denotes a ball with resp ect to the CarnotCaratheo dory

metric

Even if proving inequalities like requires many assumptions on X there are

suciently many imp ortant examples where holds Some of them will b e discussed

in the following subsections

Theorem Assume that a system of local ly Lipschitz vector elds is such that

condition is satised Fix C and p Then the space

H supports a pPoincare inequality with given and C if and only if in

e e

equality holds whenever B and u C B

Proof The lefttoright implication is easy to obtain as the function jXuj is an upp er

gradient of u for u C If u C B then we can extend u from the ball

B to a continuous function on next extending jXuj from the same ball to

by gives an upp er gradient on of the extension of u Now applying the pPoincare

inequality on B to the extended pair and passing to the limit as yields

For the righttoleft implication we have to prove that whenever g is an upp er

gradientofa continuous function uthen

Z Z

g dx ju u j dx C r

e e

B B

e e e

for all B Fix a ball B We may assume that g L B otherwise the

inequality is obvious

Since g is an upp er gradient of u on B Theorem implies that jXuj g

e e

ae in B Then by Theorem there is a sequence of functions u C B

such that ku uk kXu Xuk Thus if we pass to the limit in

k k

p p

e e

L B L B

the inequality applied to u s we obtain the pPoincare inequalityfor the pair u

jXuj This together with the estimate jXuj g yields The pro of is complete

Thereisanobvious way to dene a Sob olev space asso ciated with a system of vector

elds Namelywe dene W p as the set of those u L such that

jXuj L where Xu is dened in the sense of distributions and we equip the space

p p

with the norm kuk kXuk under which W b ecomes a Banach space

L L

According to Theorem when p one can equivalently dene the space

as the completion of C in the ab ove norm

We will return to the construction of a Sob olev space asso ciated with the system of

vector elds in Section

c X where c s are arbitrarily chosen con Proof of Theorem Let Y

j j j

stant co ecients with c

j j

Let x t be the function uniquely dened by the conditions x x and

x t Y x t The prop erties of are collected in the following lemma For

the pro of see Franchi Serapioni and Serra Cassano p or Hartman

Hille

Lemma If then there exists T such that T T

Moreover for every t T T the mapping t is biLipschitz onto

the image with the inverse t the mapping t is dierentiable ae and

x t a x t

ij ij

where is the Kronecker symbol and ja x tjC jtj with a constant C which does

ij ij

not depend neither on x nor on t T T This implies that the Jacobian of

satises

e e

J x tJ x t jJ x tjC jtj

for the given range of x and t

Let It suces to show that jXuj g ae in the theorem will follow then

by an exhaustion of the domain

Dene the directional derivativeofu in the direction of Y by the formula Yux

j ux t

The plan of the pro of of the theorem is the following In the rst step we prove that

e e e

Yu exists ae and that jYujg ae In the second step we prove that Yu is actually

the distributional derivative and in the last step weshow that by an appropriate choice

of the c s we get jYuj jXuj

e e

Step We show that Yu exists ae and jYuj g ae

e e

If Y x then Yux and hence jYuxj g x Thus it remains to prove

the inequality in the op en set where Y

Observe that the curves t x t are admissible and hence for T t t

g x t dt jux t ux t j

Thus if for given x the function t g x t is lo cally integrable then the ab ove

inequality implies that the function t ux t is absolutely continuous and

jYux tj ux t g x t

for almost all t T T

Fix x with Y x and let B x b e a suciently small ball contained

in the hyp erplane p erp endicular to Y x For a moment restrict the domain of deni

tion of to G B x T T The uniqueness theorem for ODE implies that

is onetoone on G Moreover the prop erties of collected in Lemma imply

that is Lipschitz on G and the Jacobian of G satises C jJ j C

on G provided and T are suciently small note that this is the Jacobian of a dif

ferent mapping than that in Hence jJ z j is b ounded on G Note

that jJ z j is dened almost everywhere on G This follows from the

observation that if E G jE j then by the change of variables formula

jE j jJ j and hence j E j The last observation implies also

that if weprove that some prop erty holds for almost all x t B x T T

then it is equivalent to say that the prop erty holds for almost all z G

The set G is op en and it contains x Since we can cover the set where Y

with such Gs it remains to prove that jY jg ae in G

In order to prove that for almost every z G the directional derivative Yuz

exists and satises jYuz j g z it suces to prove that for almost every x

B x the function t g x t t T T is integrable and then the claim

follows from The integrabilityfollows immediately from the estimate

Z Z Z Z

g x t dt dx g z jJ z j dz C g z dz

B x T G G

and the Fubini theorem Thus we proved that jYuj g ae in G and hence ae in

Step Now we prove that Yu Yu where Yu is the distributional derivative

dened by its evaluation on C by the formula

Z Z Z

hYui uY uY u div Y

In the pro of we will need a stronger result than just the inequality jYuj g Let

u z uz uz We claim that for every C

Z Z

u z z dz Yuz z dz

Since u Yu ae it suces to prove that lo cally the family fu g is uniformly

integrable Then the convergence will follow from Prop osition According to

the Vallee Poussin theorem see Theorem it suces to prove that there exists a

convex function F such that F F xx as x and

sup F ju j where G was dened in the rst step

Since g L G then again by Vallee Poussins theorem there is a convex

function F with growth prop erties as ab ove and such that F g Now

ju x tj jux t ux tj g x s ds

Hence Jensens inequality implies

F g x s ds F ju x tj

and thus

Z Z Z Z Z

T T t

F ju x tj dt dx F g x s ds dt dx

n n

B x T B x T t

Z Z

F g x s ds dx

B x T

Therefore

Z Z Z

sup F ju z j dz sup F ju x tj jJ x tj dt dx

G B x T

T T

Z Z

C F g x s ds dx

B x T

which yields desired uniform integrability This completes the pro of of

Now we pro ceed to prove that Yu Yu Fix an arbitrary x Note that

Yu Y u ux and hence

Z Z

e e

jhYu Yuij Yu u ux Y ju ux jjjjdiv Y j I I

First we prove that sup jhYu Yuij C x where the supremum is taken

over all C B x with kk This inequality implies that Yu Yu is a

signed Radon measure with total variation on B x equal to C x

In what follows we assume that is compactly supp orted in B x with the

supremum norm no more than As Y has lo cally Lipschitz co ecients jdiv Y j is

lo cally b ounded and hence

I C sup ju ux j

B x

The estimates for I are more dicult to handle In what follows we write u instead of

u ux and simply assume that ux We have

Z Z Z

uxYx dx lim uxx dx uxx t dx A

The change of variables x x t together with yields

Z Z

e e e e

uxx t dx ux tx J x t dx

and hence by

Z Z

ux ux t

A lim x dx lim ux txJ x t dx

t t

Z Z

e e

ux txJ x t dx Yuxx dx lim

Hence

I lim ux txJ x t dx

C lim jux tj dx C sup ju ux j

B x

This and the estimate for I yields that Yu Yu is a signed Radon measure whose

total variation on B x is estimated from ab ove by

n n

jYu YujB x C sup ju ux j

B x

This in turn implies that the measure jYu Yuj is absolutely continuous with resp ect

to the Leb esgue measure so Yu Yu L and then by the Leb esgue dierentiation

lo c

theorem

e e

jYu Yuj C lim sup ju ux j jYux Yux j lim

B x

for almost all x Thus Yu Yuand hence by Step jYuj g ae

Step Rep eating the ab ove arguments for all the rational co ecients c we con

clude that there is a subset of of full measure such that for all rational c s with

P P

c there is j c X uj g at all p oints of the set If jXuj ata given p oint

j j

then jXuj g at the p oint If jXuj then approximating co ecients c X ujXuj

j j

by rational co ecients and passing to with the accuracy of the approximation yields

jXuj j c X uj g The proofofthe theorem is complete

j j

Carnot groups

The aim of this subsection is to give a background on the socalled Carnot groups

which are prime examples of spaces that supp ort the pPoincare inequality for any

p Carnot groups are sp ecial cases of CarnotCaratheo dory spaces asso ciated

with a system of vector elds satisfying Hormanders condition that will be describ ed

in the next subsection For a more complete intro duction to Carnot groups see Folland

and Stein Chapter and also Heinonen

Before we provide the denition we need to collect some preliminary notions and

results

Let g be a nite dimensional real Lie algebra We say that g is nilpotent of step

m if for some p ositive integer m g fg g fg where g g and

m m

g g g A Lie algebra is called nilpotent if it is nilp otent of some step m A

j j

Lie group G is called nilpotent of step m if its Lie algebra is nilp otentof step m

Let V be the underlying vector space of the nilp otent Lie algebra g Dene the

p olynomial mapping V V V by the Campb ellHausdor formula

X X

n m n m

p p

X Y

p n m n m

p p

n m

i i

n m n m

p p

ad X ad Y ad X ad Y Y

where ad AB A B We adopt here the convention that if m then the term

in the sum ends with adX X Note also that if m then the term in the

sum is zero

The formal series on the right hand side of is in fact a p olynomial b ecause

the Lie algebra is nilp otent One can check that the mapping denes a group struc

ture on V with the Lie algebra g Since connected and simply connected Lie groups

with isomorphic Lie algebras are isomorphic we obtained a full description of simply

connected nilp otent Lie groups

In what follows the group identity will b e denoted by however for the group law

we use multiplicative notation xy

One can write formula in the form

X Y X Y X Y

where the dots indicate terms of order greater than or equal to Note that the map

t tX is a one parameter subgroup of V Hence the exp onential map exp g V

is identity Then one can nd a basis in V so that in the induced co ordinate system

the Jacobi matrix of the left multiplication by a V is alower triangular matrix with

ones on the diagonal Thus the Jacobi determinant equals one Hence the Leb esgue

measure is the left invariant Haar measure The same argument applies to the right

multiplication and so the Leb esgue measure is the biinvariantHaar measure

A Carnot group is a connected and simply connected Lie group G whose Lie algebra

g admits a stratication g V V V V V V fg for im Obviously

m i i i

a Carnot group is nilp otent Moreover a Carnot group is nilp otent of step m if V

Note that the basis of V generates the whole Lie algebra g Carnot groups are also

known as stratied groups

Being nilp otent Carnot group is dieomorphic to IR for some n

Let X X X form a basis of V We identify X X X with the left

k k

invariantvector elds

The following result is due to Chow and Rashevsky For mo dern

pro ofs see Bellache Gromov Herman Nagel Stein and Wainger

Strichartz Varop oulos SaloCoste and Coulhon

Prop osition The CarnotCaratheodory distance associated with the basis X

X X of V is a metric ie every two points of the Carnot group can beconnected

by an admissible curve

The aim of this subsection is to prove that a Carnot group with the ab ove Carnot

Caratheo dory metric supp orts the pPoincare inequalities for all p see

Theorem This is a sp ecial case of Jerisons result see Theorem that will

be describ ed in the next subsection

By G we will denote a Carnot group of step m and will be the Carnot

Caratheo dory metric asso ciated with the basis X X of V

As the CarnotCaratheo dory metric is not given in an explicit form it is quite

dicult to handle it Therefore it is convenienttointro duce new distances that can b e

dened explicitly and that are equivalent to the CarnotCaratheo dory metric

A Carnot group admits a oneparameter family of dilations that we next describ e

For X V and r we set X r X This extends to a linear map that is an

i r

automorphism of the Lie algebra g This in turn induces an authomorphism of the Lie

group via the exp onential map

Observe that the metric has the two imp ortant prop erties of b eing left invariant

and commutative with in the sense that x y rx y

r r r

A continuous homogeneous norm on G is a continuous function j j G

that satises jx j jxj j xj r jxj for all r and jxj if and only if

One such homogeneous norm is given by jxj x

Prop osition Let jj be a continuous homogeneous norm Then the fol lowing

results hold

There exist constants C C such that C kxkjxj C kxk for jxj

Here kk denotes a xed Euclidean norm

The distance x y jx y j is a quasimetric ie it has al l the properties of

metric but the triangle inequality that is replaced by a weaker condition there is

a constant C such that for al l x y z G

x y C x z z y

Bal ls B x r fy x y r g are the left translates of B r by x and

B r B

The number Q j dim V wil l be cal led the homogeneous dimension It

Q Q

satises j E j r jE j and hence jB x r j Cr for al l x G and al l r

where jE j denotes the Lebesgue measure of the set E

Any two continuous homogeneous norms are equivalent in the sense that if jj

is another continuous homogeneous norm on G then there exist C C such

that C jxj jxjC jxj for al l x G

For a pro of see Folland and Stein Chapter Anyway the pro of is easy and it

could be regarded as avery go o d exercise

In the literature the concept of a homogeneous norm is dened as ab ove but with

the additional prop erty of b eing C smo oth on G nfg This prop erty is irrelevantto

us Thus we delete it and add the adjective continuous to indicate that we do not

assume smo othness

To give an explicit example of a continuous homogeneous norm note that any ele

ment x V can b e represented as x x where x V Fix an Euclidean norm

j j j

kk in V Then

jxj kx k

is acontinuous homogeneous norm on G after identication of G with V

The continuous homogeneous norm induced by the CarnotCaratheo dory metric

x x satises with C For general continuous norms we only have

C See also Hebisch and Sikora for a construction of a homogeneous norm

ie smo oth on G nfgwithC

Mitchell proved that the Hausdor dimension of a Carnot group is equal to

its homogeneous dimension see also p This dimension in general is larger

than the Euclidean dimension of the underlying Euclidean space This shows that the

CarnotCaratheo dory geometry is pretty wild and the metric is not equivalent to any

Riemannian metric

It is an exercise to show that inequality of the ab ove prop osition implies that for

every b ounded domain G there are constants C C such that

C kx y k x y C kx y k

whenever x y Note that inequality along with Lemma imply that every

two points can be connected by a geo desic the shortest admissible curve

So far wehave not given any examples of the Carnot group Let us ll the gap right

now

Example The most simple nontrivial example of a Carnot group is the Heisen

berg group IH C IRwith the group law

z t z t z z t t Imz z

The basis consisting of the left invariantvector elds X Y Z such that X x

Y y T t is given by

X y Y x T

x t y t t

Note that X Y T and all the other commutators are trivial Thus the Lie

algebra is stratied h V V with V spanfX Y g and V span fT g The

CarnotCaratheo dory metric is dened with resp ect to the vector elds X Y The

group IH is a nilp otent group of step and its homogeneous dimension is The

family of dilations is given by z t rz r t for r Moreover the function

jz tj t jz j is a homogeneous norm

The following theorem states that a Carnot group supp orts a Poincare inequality

This is a corollary of a much more general theorem of Jerison see Theorem For

completeness we provide a clever pro of due to Varop oulos see also page

Prop osition Any Carnot group equipped with the Lebesgue measure and the

CarnotCaratheodory metric supports a Poincare inequality

Proof Let G b e a Carnot group with the CarnotCaratheo dory metric that we denote

by Let u g be a pair of a continuous function and its upp er gradient It suces to

prove that

Z Z

jux u j dx Cr g x dx

B B

on every ball of radius r Obviously we can assume that the ball B is centered at Set

jz j z and let jz j G be a geo desic path that joins with z Observe

that s x s is the shortest path that joins x with xz Hence

jz j

jux uxz j g x s ds

This and the left invariance of the Leb esgue measure yields

Z Z Z

jux u j dx jux uy j dy dx

jB j

B B B

Z Z

x xz jux uxz j dz dx

B B

jB j

G G

Z Z Z

jz j

x xz g x s ds dx dz

B B z

jB j

G G

Invoking the rightinvariance of the Leb esgue measure we obtain

Z Z

x xz g x s dx g d

B B z

B s

Bz s

G G

z g d

Here we denote by Bh the right translation of B by h The ab ove inequality requires

some explanations If the expression under the sign of the middle interval has a nonzero

value then x s yz s for some x y B Hence z x y B Thus

z z

x s lies on a geo desic that joins x with y and so x y x y

x y

which together with the triangle inequality implies B and hence follows Now

Z Z Z Z

jz j

jux u j dx z g d ds dz

B B

jB j

B G B

Z Z

jz jg d dz

jB j

B B

Cr g d

The pro of is complete

Remarks The ab ove pro of easily generalizes to more general unimo dular groups

see page

Applying Theorem to inequalitywe conclude that the ball B on the right

hand side can b e replaced by B and moreover the exp onent on the left hand side can

be replaced by QQ where Q is the homogeneous dimension of the group This

inequality in turn implies the isop erimetric inequality Such an isop erimetric inequality

was proved rst in the case of the Heisenb erg group byPansu and in the general

case of the Carnot groups by Varop oulos

For a more complete treatment of Sob olev inequalities on Lie groups with the

CarnotCaratheo dory metric see Gromov and Varop oulos SaloCoste and Coul

hon and also Folland Nhieu

Hormander condition

Denition Let IR be an op en connected set and let X X X be vector

elds dened in a neighb orho o d of real valued and C smo oth We say that

these vector elds satisfy Hormanders condition provided there is an integer p such

that the family of commutators of X X X up to the length p ie the family

of vector elds X X X X X X X i k span

k i i i i i j

the tangent space IR at every point of

The denition easily extends to smo oth manifolds but for simplicitywe will consider

the Euclidean space only

x where k is As an example take the vector elds X x X x

a p ositive integer These two vector elds do not span IR along the line x

However X X and commutators of the length k do

Another example is given by vector elds on a Carnot group Namely if G is a

Carnot group see the previous subsection with the stratication g V V of

its Lie algebra then the left invariantvector elds asso ciated with a basis of V satisfy

Hormanders condition

The ab ove condition was used by Hormander in his celebrated work on hy

p o elliptic op erators see also Bony Chemin and Xu Feerman and Sanchez

Calle Hormander and Melin Jerison Morbidelli Nagel Stein

and Wainger Rothschild and Stein SanchezCalle Varop oulos Salo

Coste and Coulhon Related references will also b e given in Section

As usual the CarnotCaratheo dory distance asso ciated with a family of vector elds

satisfying Hormanders condition will b e denoted by

The following result provides the full version of the theorem of Chow and

Raschevsky whose sp ecial case was discussed earlier see Prop osition For the

pro of see the references given there In some more simple settings the theorem was

proved earlier by Caratheo dory

Theorem Let an open and connected set IR and a system of vector elds

satisfying Hormanders condition in be given Then any two points in can be

connectedbya piecewise smooth admissible curve and hence the CarnotCaratheodory

distance is a metric

Nagel Stein and Wainger studied the geometry of CarnotCaratheo dory spaces

in detail and in particular they gave a more quantitativeversion of ChowRaschevskys

theorem Let us quote some of their results

In what follows B x r will denote a ball with resp ect to the metric

Theorem Let X X be a system of vector elds satisfying Hormanders

condition as above and let be the associated CarnotCaratheodory metric Then for

every compact set K there exist constants C and C such that

C jx y j x y C jx y j

for every x y K Moreover there are r and C such that

jB x r j C jB x r j

whenever x K and r r

Here as usual jB j denotes the Leb esgue measure In the previous subsection we proved

the theorem in the sp ecial case of a Carnot group The general case is however much

more dicult see also Gromov and Varop oulos SaloCoste and Coulhon

Section IV Estimate has b een obtained indep endently by Lanconelli

n n

Assume for a moment that IR If IR is bounded with resp ect to the

Euclidean metric then by it is also b ounded with resp ect to However if is

bounded with resp ect to thenitneednotbeboundedwith resp ect to the Euclidean

metric Indeed if one of the vector elds is x x then the CarnotCaratheo dory

distance to innity is nite b ecause of the rapid growth of the co ecient Hence in

general holds only for r r for some suciently small r and r cannot be

replaced with diam even if diam aswas required for the measure in

the denition of doubling in

Prop osition is a sp ecial case of the following Poincare inequality of Jerison

see also Jerison and SanchezCalle and Lanconelli and Morbidelli

Theorem Let X X be a system of vector elds satisfying Hormanders

condition in Then for every compact set K there are constants C and

r such that for u Lip B

Z Z

ju u j dx Cr jXuj dx

B B

whenever B is a bal l centered at K with radius rr

In fact Jerison proved the inequality with the L norms on b oth sides but the same

argument works with the L norm Then Jerison proved that one can replace the ball

B on the righthandsideofwithB As wehave already seen this can b e done in

amuch more general setting see Section

Further generalizations

The results of the previous two subsections concern Poincare inequalities for smo oth

vector elds satisfying Hormanders condition It is a dicult problem to nd a large

class of vector elds with Lipschitz co ecients suchthatthePoincaretyp e inequalities

hold on the asso ciated CarnotCaratheo dory spaces The lack of smo othness do es not

p ermit one to use a Hormander typ e condition There are few results of that typ e

see Franchi Franchi Gutierrez and Wheeden Franchi and Serapioni

Franchi and Lanconelli Jerison and SanchezCalle It seems that Franchi

and Lanconelli were the rst to prove a Poincare typ e inequality for a Carnot

Caratheo dory space They probably also were the rst to prove estimates of the typ e

as in Theorem

Graphs

Let G V E be a graph where V is the vertex set and E the set of edges We say

that x y V are neighb ors if they are joined by an edge we denote this by x y

Assume that the graph is connected in the sense that anytwovertices can b e connected

by a sequence of neighbors We let the distance between two neighb ors to be This

induces a geo desic metric on V that we denote by The graph is endowed with the

counting measure the measure of a set E V is simply the number V E of elements

of E For a ball B B x r we use also the notation V B V x r Wesay that G is

local ly uniformly nite if d sup dx where dxisthe numb er of neighbors

of x The length of the gradient of a function u on V atapoint x is

jr ujx juy uxj

y x

Many graphs have the following two prop erties

The counting measure is doubling ie

V x r C V x r

for every x V and r

The pPoincare inequality holds ie there are constants C and such

that

X X

jux u jC r jr uj x

B P G

V B V B

xB xB

for any ball B and any function u V IR

Observe that the doubling condition implies that the graph is lo cally uniformly nite

The Euclidean or more generally the upp er gradients have the truncation prop erty

Unfortunately the truncation prop erty is no longer valid for the length of the gradienton

j is not supp orted on the set ft u t g a graph This is b ecause in general jr u

However intuition suggests that jr uj should still have prop erties similar to those of

a gradient with the truncation prop erty

If v Lip IR and p then

X X

p p p

j jrv j jrv j jrv

k k

f v g

k k

almost everywhere It turns out that inequality is satised also by the length of

the gradient on a graph More precisely we have the following estimate

Lemma Let G be local ly uniformly nite ie d sup dx If v V

IR and p then

r v x C p djr v j x

G G

for each x V

Proof Fix x V and let

v xmaxfv w w x g

v x minfv w w x g

Note that jr v jx jv x v xj Assume for simplicitythat v x the case

G M m m

v x follows by the same argument Let j ZZ be the least integer and i ZZ

the largest integer such that

j i

v x v x

M m

We have

k k i j

v x jv x v xj v x

m M m M

k i

Hence

k i p p j p

A A

j v xj jr v j x C jv x j

m G M

k i

j p p kp i p

C jv x j j v xj

M m

k i

j k i

p p p

j x jr v jr v jr v j x j x C

j i

G G G

p p p

d d d

k i

C p

jr v j x

k i

C p

jr v j x

The pro of is complete

The inequality of the lemma is a go o d substitute for the truncation prop erty it

allows one to mimic the pro ofs of Theorems and We will generalize Corol

lary This result deals with sharp inequalities with integrals on the dierent sides

of the inequality taken over the same domain As pointed out in Section a Poincare

inequality do es not in general guarantee that one could use balls of the same size on

the dierent sides of the inequality We describ ed a sucient condition in terms of the

geometry of balls that in particular holds for the CarnotCaratheo dory metrics As

is a geo desic metric it should come as no surprise that we can reduce the size of in

down to

Theorem Assume that the counting measure is doubling and that for some con

stants C s

V x r r

whenever B x r B x r Suppose that each function u V IR satises the

pPoincare inequality with a xed p

If p s then there is a constant C such that

spsp p

X X

spsp p

jux u j jr uj x Cr

B G

V B V B

xB xB

for any bal l B of radius r and any function u V IR

If p s then there are constants C C such that

C V B jux u j

C exp

V B diam B kr uk

L B

for any bal l B of radius r and any function u V IR

If ps then there is C such that

sp sp p

jux uy j Cx y r V B kr uk

G L B

for al l x y B where B is an arbitrary bal l of radius r and for any function

u V IR

The constants C C C depend on p s C C and C only

d b P

Remark If sps p then we have to replace u by u in where B

B B

Proof The pro of involves arguments similar to those used in the previous sections

so we sketch the main ideas only leaving details to the reader

First of all under the lo cal niteness of the graph one may assume that r as

for r wehave a nite collection of nonisometric balls only Wefollow the line of

ideas from Section Given a ball B x r and a p oint x B x r we join x to x by

achain x x x of length less than r of vertices If we trace along the chain for l

steps with l the least integer larger or equal to then B x B x r Following

the chain towards x wemay construct a chain B B x r B B B x

k l

of balls as in the C M condition of Section Next

l k

X X

jux u j jux ux j ju ux j ju u j

B i i l B B

B x

i i

X X

jr ujx jr ujy

G i G

y B x

X X

jr uj y C r

G i

V B

y B

We employed here the observation that jux ux j jr ujx Inequality

i i G i

is a go o d substitute for as jr ujy equals to the pro duct of the radius and

the L average of jr uj over the ball B y

Now assume that ps Write sps pp Using aversion of Theorem as

in Section we conclude the weak typ e inequality

p p

V fx B jux u j tgt

p p

sup Cr jr uj x

V B V B

To obtain the desired strong typ e inequality one reasons as follows

Dene v as in the pro of of Theorem with replaced by B It suces to

prove suitable L estimates for v and v In what follows v denotes either v or v

We have

p p

V fx B v tgt

p t p

Cr jr v j sup

V B V B

and hence

kp k

X X

V fx B v g

v x

V B V B

xB k

p p

X X

p p

jr v j x Cr

V B

p p

jr v j x Cr

V B

p p

Cr jr uj x

V B

If p s then the metho d describ ed ab ove provides us with a chains that are

suciently go o d to mimic the pro of of Theorem

Once we have go o d chains also the Holder continuity with the same balls on b oth

sides follows when ps

The pro of is complete

Remarks The doubling prop erty and the Poincare inequality are very

imp ortant in the p otential theory on graphs Indeed indep endently Delmotte

Holopainen and Soardi and Rigoli Salvatori and Vignati proved

that and imply the socalled Harnack inequality for pharmonic functions As

a consequence they concluded a Liouville typ e theorem stating that every b ounded

pharmonic function on G is constant Recall that u V IR is harmonic if it

uy for all x V For a satises the mean value property ie ux dx

y x

denition of pharmonic functions with p see eg The pro of of the Harnack

inequality employs the Sob olevPoincare inequality as it relies on the Moser iteration

As we have already seen prop erties and imply the Sob olevPoincare inequality

Pap ers related to the Harnack inequalities on graphs include Chung Chung and

Yau Lawler Merkov Rigoli Salvatori and Vignati Schinzel

Zhou

The Moser iteration was originally employed in the setting of elliptic and parab olic

equations see the next section

There are many examples of graphs for which both prop erties and are

satised A very nice example is given by a Cayley graph asso ciated with a nitely

generated group We say that the group G is nitely generated if there is a nite set

f g such that every element g G can be presented as a pro duct g

i i

Then the vertex set of the Cayley graph is the set of all elements of G and

two elements g g G are connected by an edge if g g for some generator

Thus wemay lo ok at nitely generated groups as geometric ob jects This p oint of view

has b een intensively used after Milnors pap er

We say that the group is of p olynomial growth if V r Cr for all r and

some C One of the most b eautifully results in the area is due to Gromov He

proved that the group is of p olynomial growth if and only if it is virtually nilp otent

and hence by the theorem of Bass V r r for some p ositiveinteger d

Thus if the group is of p olynomial growth then it satises the doubling prop erty

It is also known that it satises the Poincare inequality

Prop osition If G is a nitely generated group of polynomial growth V r r

d positive integer then there is a constant C such that

X X

jux u j Cr jr uxj

B G

V B V B

xB xB

for every bal l B X

The reader mayprove the prop osition as an easy exercise mimicing the pro of of Theo

rem

Other examples of graphs with prop erties and can be found in Holopainen

and Soardi Coulhon and SaloCoste SaloCoste

The analysis on graphs is also imp ortant in the study of op en Riemannian man

ifolds b ecause roughly sp eaking one can asso ciate with given manifold a graph with

similar global prop erties This metho d of discretization of manifolds has b een in active

use after the pap ers of Kanai and Mostow Related references include

Auscher and Coulhon Coulhon Coulhon and SaloCoste

Chavel Coulhon Delmotte Holopainen Holopainen

and Soardi Soardi Varop oulos and Varop oulos SaloCoste

and Coulhon

Applications to PDE and nonlinear p otential

theory

The results presented in the pap er directly apply to the regularity theory of degen

erate elliptic equations asso ciated with vector elds Below we describ e some of the

applications

Admissible weights

n m m

Let A A A IR IR IR be a Caratheo dory function satisfying the

growth conditions

p p

jAx j C xj j Ax C xj j

where p C C are xed constants and L IR We will

lo c

denote by d dx the measure with the density

Given A we consider the equation

X A x X uX u

j m

where X X X is a family of vector elds with lo cally Lipschitz co ecients

R R

in IR Recall that X denotes the formal adjoint of X that is X uv uX v for

j j

all u v C

The theory of nonlinear equations of the typ e esp ecially when X is a system

of vector elds satisfying Hormanders condition is an area of intensive research see

eg Buckley Koskela and Lu Cap ogna Cap ogna Danielli and Garofalo

Chernikov and Vo dopyanov Citti Citti and Di Fazio Citti

Garofalo and Lanconelli Danielli Garofalo and Nhieu Franchi Gutierrez

and Wheeden Franchi and Lanconelli Franchi and Serapioni Garofalo

and Lanconelli Garofalo and Nhieu Ha jlasz and Strzelecki Jerison

Jerison and Lee Jost and Xu Lu Marchi

Vo dopyanov and Markina Xu Xu and Zuily The ab ove

pap ers mostly deal with the nonlinear theory References to the broad literature on the

linear theory can be found in these pap ers

Equation is a generalization of the classical weighted pharmonic equation

Indeed if X r and A xj j we get the equation

div xjruj ru

In what follows we assume that the condition is satised ie we assume that the

CarnotCaratheo dory distance is a metric such that the identity map is a homeo

morphism b etween the Euclidean metric and

We call u a weak solution of if

A x X u X x dx

j j

for any C IR We assume that the weak solution belongs to the weighted

Sob olev space W IR dened as the closure of C functions in the norm

Z Z

p p

kuk juxj x dx jXuxj x dx

n n

IR IR

Already in the classical case ie when Xu ru one has to put many additional

conditions on the weight in order to have a reasonable theory

The rst condition concerns the denition of the Sob olev space One needs the

socalled uniqueness condition which guarantees that the gradient X is well dened

in the Sob olev space asso ciated to X Later we will clarify this condition

The regularity results for solutions to like Harnack inequality and Holder

continuity are usually obtained via the Moser iteration technique For that the essential

assumptions are a doubling condition on with resp ect to the CarnotCaratheo dory

metric the Poincare inequality

Z Z

p p

ju u j d jXuj d Cr

e e

B B

for all smo oth functions u in a metric ball B and a Sob olev inequality

Z Z

q p

Cr d d juj jXuj

e e

B B

with some qp for all smo oth functions u with compact supp ort in a metric ball B

Given the ab ove assumptions one can mimic the standard Moser iteration technique

replacing Euclidean balls by metric balls This leads to the Harnack inequality which

states that if u is a p ositive solution to on B then

u u C inf sup

where the constant C do es not dep end on B Then the iteration of the Harnack

inequality implies that each weak solution to is lo cally Holder continuous with

resp ect to and hence if the condition is satised lo cally Holder continuous

with resp ect to the Euclidean metric

The fact that the ab ove conditions are essential for the Moser iteration was observed

rst byFab es Kenig and Serapioni They considered the classical linear setting

X r p

It seems that Franchi and Lanconelli were the rst to apply the Moser tech

nique for the CarnotCaratheo dory metric as ab ove Then the idea was extended

by many authors to more dicult situations see eg Cap ogna Danielli and Garo

falo ChernikovandVo dopyanov Franchi Franchi and Lanconelli

Franchi Lu and Wheeden Franchi and Serapioni Jerison Lu

There are moreover many other related pap ers

SaloCoste and Grigoryan indep endently realized that in certain set

tings a Poincare inequality implies a Sob olev inequality and hence one can delete

assumption as it follows from This result was extended then to more general

situations by several authors Biroli and Mosco Maheux and SaloCoste

Ha jlasz and Koskela Sturm Garofalo and Nhieu

The result presented below Theorem is in the same spirit This is a general

ization of a result of Ha jlasz and Koskela

The following denition is due to Heinonen Kilp elainen and Martio when

X r and due to Chernikov and Vo dopyanov in the case of general vector

elds

We say that L IR ae is padmissible p if the measure

lo c

dened by d x dx satises the following four conditions

e e e

Doubling condition B C B for all metric balls B IR

Uniqueness condition If is an op en subset of IR and C is a sequence

R R

p p p

suchthat j j d and jX v j d where v L then v

i i

Sob olev inequality There exists a constant k such that for all metric balls

e e

B IR and all C B

Z Z

k p p

kp p

jj d C r jXj d

e e

B B

e e

Poincare inequality If B IR is a metric ball and C B then

Z Z

p p p

j j d C r jXj d

e e

B B

One can easily mo dify the ab ove denition and consider vector elds dened in an

op en subset of IR with the estimates in the ab ove conditions dep ending on compact

subsets of However for clarity we assume the global estimates We do not care

to present various results in their most general form We aim to present the metho d

Various generalizations are then obvious

The uniqueness condition guarantees that any function u L IR that b elongs

to W IR has a uniquely dened gradient Xu as the limit of gradients Xu of

smo oth functions u which converge to u in the Sob olev norm If the uniqueness

condition were not true then we would nd u C such that u in L and

k k

Xu v inL Then the zero function would have at least two gradients and

IR v ie and vwould be two distinct elements in W

n n

Theorem Let L IR and let X be a system of vector elds in IR

lo c

satisfying condition Then the weight is padmissible p if and

only if the measure associated with is doubling with respect to the metric ie

e e e

B C B for al l metric bal ls B IR and there exists such that

Z Z

ju u j d Cr jXuj d

e e

whenever B IR is a metric bal l of radius r and u C B

Proof The necessity is obvious Now we prove the suciency First note that the

uniqueness of the gradient was recently proved byFranchi Ha jlasz and Koskela

Corollary

Next by Corollary we conclude the Sob olevPoincare inequality

Z Z

p p

j j jXj d d Cr

e e

B B

for all C B with some p p remember that the doubling condition implies

with s log C For our purp ose the exact value of p is irrelevant It is only

imp ortant that p p This and the Holder inequality implies the Poincare inequality

Now we are left with the Sob olev inequality Since p p we have p kp for

B we have some k For C

Z Z

k p k p

kp kp

j j jj d j j d

e e

B B

e e

B B

The Sob olevPoincare inequality provides us with the desired estimate for the rst

summand on the right hand side Now it suces to estimate j j The Poincare

inequality applied to the ball B gives

Z Z

p p

jXj d j j d Cr

e e

B B

and when applied to the ball B gives

Z Z Z

p p p

j j d j j d C r jXj d

e e

B B

e e

B B

Cr jXj d

Thus

Z Z

p p

j j d j j d

e e e

B B B

e e

B B

Cr jXj d

In the pro of of the equalitywe employ the fact that is supp orted in B and the inequal

ity follows from the triangle inequality and inequalities and It follows from the

e e

doubling prop erty and the geometry of metric balls in IR that B B C

and hence

jXj d j C r j

The pro of is complete

Sob olev emb edding for p

The classical Sob olevPoincare inequality

Z Z

p p

dx C ju u j jruj

B B

holds when p n It is easy to see that it fails when p and even a

weaker version of the Poincare inequality fails for the range p For an explicit

example see Buckley and Koskela

However Buckley and Koskela and in a more general version Buckley Koskela

and Lu proved that if u is a solution to the equation div Ax X u in a John

domain with resp ect to the CarnotCaratheo dory metric then u satises a Sob olev

Poincare inequality for any ps where s is given by condition

As we will see one of the results of the pap er Theorem which states that for

any p s a pPoincare inequality implies a Sob olevPoincare inequality can be

regarded as an abstract version of the ab ove result In particular this gives a new pro of

of the result of Buckley Koskela and Lu

More precisely assume that X X X are lo cally Lipschitz vector elds in

IR Assume that the asso ciated CarnotCaratheo dory metric satises condition

the Leb esgue measure is doubling with resp ect to the CarnotCaratheo dory distance

e e e

ie jB j C jB j for all metric balls B IR and that condition is satised

In addition we assume that the Poincare inequality is satised ie there is C

and such that

Z Z

jXuj dx ju u j dx C r

e e

B B

e e

for all metric balls B IR and all u C B

n m m

Let A IR IR IR be a Caratheo dory function such that

q q

jAx j C j j Ax C j j

where q is given Observe that in contrast with the previous section we do

not allow the weight

The following result is a variantof the result of Buckley Koskela and Lu

Theorem Let IR be a John domain with respect to the Carnot

Caratheodory metric Then for any p s there is a constant C such that

if u is a solution to equation div Ax X u in then

Z Z

p p

inf ju cj dx C diam jXuj

cIR

The constant C depends on n p s C C C C C and C only

b d P J

Proof Let u be a solution to div Ax X u in The rst fact we need is that the

gradient jXuj of the solution u satises a weak reverse Holder inequality This is well

known However for the sake of completeness we provide a pro of

Given a metric ball B let b e a cuto function such that j

R R R

outside B and jX jR Using the distance function with resp ect to we

R R

easily construct a cuto function with the metric Lipschitz constant R Then the

estimate jX jR follows from Corollary

Now taking a test function u u where B is any metric ball and

is the asso ciated cuto function we conclude from a standard computation the

Cacciopp oli estimate

Z Z

q q

jXuj ju u j

e e

B B

Then we estimate the righthandsideby the Sob olevPoincare inequality and conclude

e e

that there is p q such that for all metric balls B with B

Z Z

q p

C jXuj dx jXuj dx

B B

This inequalityisknown under the name weak reverse Holder inequality

It is well known that the weak reverse Holder inequality has the selfimproving

e e

prop erty if inequalityholds for some pq and all B with B then for

e e

any p q there isanew constant C such that holds for any B with B

see Lemma This together with the Poincare inequalityshows that the pair

u g satises a pPoincare inequalityin for any p Hence the claim follows from

Theorem The pro of is complete

App endix

Here we collect the results in the measure theory that are needed in the pap er All

the material is standard Since we could not nd a single reference that would cover

the material we need wehave made all the statements precise and sometimes wehave

even given pro ofs Go o d references are Federer Mattila and Simon

In the app endix we do not assume that the measure is doubling

Measures

Throughout the pap er by a measure we mean an outer measure and by a Borel measure

an outer Borelregular measure ie such a measure on a metric space X d that all

Borel sets are measurable and for every set A there exists a Borel set B such that

A B and A B In the case of a Borel measure we also assume that the

measure of each ball is strictly p ositiveand X U where U are op en sets with

j j

Note that if the space X is lo cally compact separable and K for every

compact set K then X can be written as a union of a countable family of op en sets

with nite measure

Theorem Suppose that is a Borel measure on X d Then

A inf U

U A

U open

for al l subsets A X and

A sup C

C A

C cl osed

for al l measurable sets A X

For the pro of see Theorem and Section Theorem or

Theorem

If the space is lo cally compact and separable the supremum over closed sets in the

ab ove theorem equals to the supremum over compact sets

As a corollary to the ab ove theorem we obtain the following well known result

Theorem If is a Borel measure on a metric space X d then for every

p continuous functions are dense in L X

Proof Simple functions are dense in L X see Theorem so it suces to

prove that characteristic functions can be approximated by continuous functions Fix

If A X is measurable A then there exists a closed set C and an

op en set U such that C A U U n C Now by Urysohns lemma there

exists a continuous function on X such that j and j

X nU

Then obviously k k as This completes the pro of

A p

In order to have a variaty of Borel measures one usually assumes that the space

be lo cally compact In the denition of the doubling measure one do es not assume

anything ab out the metric space However as we will see the existence of a doubling

measure is such astrong condition that the space is almost lo cally compact

Wesay that a subset A of a metric space X disannet if for every x X there

is y A with dx y A metric space X d is called total ly bounded if for each

there exists a nite net

The following two lemmas are well known

Lemma A metric space X d is compact if and only if it is complete and total ly

bounded

Lemma Every metric space is isometric to a dense subset of a complete metric

space

The rst lemma follows from a direct generalization of the pro of that every b ounded

sequence of real numb ers contains a convergent subsequence while the second lemma

follows by adding the abstract limits of Cauchy sequences to the space

Theorem If a metric space X d admits a Borel measure which is local ly

uniformly positive in the sense that for every bounded set A X and every

inf B x

then X d is isometric to a dense subset of a local ly compact separable metric space

Proof The fact that X is a union of countably many op en sets of nite measure and

imply that X can be covered by balls X B with B

j j

According to Lemma and Lemma it suces to prove that for every j

and every there is a nite net in B This however easily follows from

and the condition B The pro of is complete

It is of fundamental imp ortance to note that the doubling condition implies lo cal

uniform p ositivityofthe measure as follows from the following result

Lemma Let be a Borel measure on a metric space X Assume that is dou

bling in the fol lowing sense on a bounded subset Y X there is a constant C

such that

B x r C B x r

whenever x Y and r diam Y Then

s s

B x r diam Y Y r

for s log C x Y and r diam Y

The ab ove lemma together with Theorem shows that doubling spaces are isometric

to dense subsets of lo cally compact separable metric spaces The anologous result holds

also when the measure is doubling on some op en set only Note that a doubling measure

is nite on b ounded sets

The ab ove remark together with the following result shows that a doubling measure

can be extended to a doubling measure on the larger lo cally compact space

Prop osition Let Y X be a dense subset of a metric space X d Let be

a Borel measure on Y d nite on bounded sets Then there exists a unique Borel

measure on X d such that

U U Y

for every open set U X Moreover if is doubling on Y d then is doubling on

X d with the same doubling constant

Proof Set A inf B Y for an arbitrary set A X One easily

B A B Borel

veries that is a Borel measure on X d This proves the existence of the measure

The uniqueness follows form Theorem

Assume now that is doubling Then obviously is doubling with the same

doubling constant on all balls centered at Y Since any ball in X can b e approximated

by balls centered at Y the result follows

Remark If we removed the assumption that be nite on b ounded sets Y would

still have the prop erty Y U where the sets U are op en with U

j j j

However then this prop ertywould not necessarily b e true for For example let Y be

the complementofaCantor set in and X Then Y consists of countable

many intervals Equip Y with a measure so that the measure of each of the intervals

is Then X cannot be decomp osed into a countable number of op en sets with nite

measure

Uniform integrability

In this section is an arbitrary measure on a set X

Assume that X We say that a family fu g of measurable functions

on X is uniformly integrable if

lim sup ju j d

The following theorem is due to Vallee Poussin For a pro of see Dellacherie and Meyer

or Rao and Ren

Theorem Let be a measure on a set X with X and let fu g be a

family of measurable functions Then the fol lowing two conditions are equivalent

The family fu g is uniformly integrable

There exists a convex smooth function F such that F

F xx as x and

sup F ju j d

The following well known result is a very useful criteria for convergence in L

Prop osition Let X If u are uniformly integrable on X and u u

n n

ju uj d ae then

Proof It follows directly from Egorovs theorem and the denition of uniform inte

grability that the sequence u is a Cauchy sequence in the L norm and hence u

n n

converges to u in L The pro of is complete

p p

L and L spaces

In the following two theorems is an arbitrary nite measure on X The rst result

is known as Cavalieris principle

Theorem If p and u is measurable then

Z Z

p p

t juj t dt juj d p

The claim follows from Fubinis theorem applied to X

We say that a measurable function u b elongs to the Marcinkiewicz space L X if

there is msuch that

juj t mt for all t

p p

If u L X then with m juj d is known as Chebyschevs inequality so

p p

L X L X The converse inclusion do es not hold However the following well

known result holds

p q

Theorem If X then L X L X for al l q p Moreover if

u satises then

q p q

X kuk

L X

p q

Proof Fix t Using Theorem and the estimates juj t X for

t t and juj t mt for tt we get

Z Z Z

q q p

q p q q

t dt t X juj d q t t X dt m

p q

X t

p p

Then inequality follows by cho osing t qmp q X

Covering lemma

Theorem r covering lemma Let B be a family of bal ls in a metric space

X d with sup fdiam B B Bg Then there is a pairwise disjoint subcol lection

B B such that

B B

If X d is separable then B is countable and we can represent B as a sequence

B fB g and so

B B

See Federer Simon Theorem or Ziemer Theorem

for a clever pro of

Maximal function

Assume that the measure is doubling on an op en set X The following theorem

is aversion of the well known maximal theorem of Hardy Littlewood and Wiener

Theorem Maximal theorem If X and are as above and the maximal

function M u is dened as in the introduction then

juj d for t and fx M ux tg Ct

p p

kM uk C kuk for p

L L

In the rst inequality the constant C depends on C only while in the second one it

depends on C and p

For a pro of in the case of Leb esgue measure see Stein Chapter We assume that

the reader is familiar with that pro of and we showhow to mo dify the argument in order

to cover our setting It suces to prove one then pro ceeds as in Inequality

would follow from this inequality for the restricted maximal function M u provided

we prove it with a constant C that do es not dep end on R To this end note rst

that the doubling condition implies that is separable and hence the second part of

Theorem applies Then the argument from the case of the Leb esgue measure

works without any changes We had to work with the restricted maximal function in

order to know that obtain a suitable covering consisting of balls with radii less than R

if we did not have the upp er b ound for the radii we could not apply Theorem

We will also need a more general result For c and x dene F x as the

family of all measurable sets E suchthatE B x r and B x r cE for

some r Then we dene a new maximal function as follows

juj d ux sup M

E F x

Obviously M u cM u and thus we obtain as a corollary to Theorem the

following result

Corollary Theorem holds with M replaced by M The only dierence

is that now the constants C in Theorem depend also on c

Leb esgue dierentiation theorem

We say that a sequence of nonempty sets fE g converges to x if there exists a

sequence of radii r such that E B x r and r as i

i i i i

Theorem Let be doubling on X and u L Then for ae

lo c

x we have

uy dy ux lim

B xr

Moreoverifwexc then for ae x and every sequence of sets E F x

i c

i that converges to x we have

lim uy dy ux

See Chappter for a pro of in the case of the Leb esgue measure in IR The

same argument works also in our setting as it only relies on two facts the weak typ e

inequality for the maximal function see Theorem and Corollary and the

density of continuous functions in L see Theorem

Let b e doubling on X Given u L it is often convenient to identify

lo c

u with the representative given everywhere by the formula

ux lim sup uy dy

B xr

Theorem shows that the taking the limit ab ove only mo dies u on a set of

measure zero We say that x is a Lebesgue point of u if

lim juy uxj dy

B xr

where uxisgiven by It follows from Theorem that almost all p oints of

are Leb esgue points of u Observe that if x is a Leb esgue point of u then b oth

and are true

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